Ch 6.4 - 33E

In Problems 1–6 use (1) to find the general solution of the given differential equation on \((0, \infty)\). \(x^{2} y^{\prime \prime}+x y^{\prime}+\left(x^{2}-\frac{1}{9}\right) y=0\) Text Transcription: (0,infty) x^2y^prime\prime+xy^prime+(x^2-frac19)y=0

In Problems 1–6 use (1) to find the general solution of the given differential equation on \((0, \infty)\). \(x^{2} y^{\prime \prime}+x y^{\prime}+\left(x^{2}-1\right) y=0\) Text Transcription: (0,infty) x^2y^prime\prime+xy^prime+(x^2-1)y=0

In Problems 1–6 use (1) to find the general solution of the given differential equation on \((0, \infty)\). \(x y^{\prime \prime}+y^{\prime}+x y=0\) Text Transcription: (0,infty) xy^prime\prime+y^prime+xy=0

In Problems 1–6 use (1) to find the general solution of the given differential equation on \((0, \infty)\). \(4 x^{2} y^{\prime \prime}+4 x y^{\prime}+\left(4 x^{2}-25\right) y=0\) Text Transcription: (0,infty) 4x^2y^prime\prime+4xy^prime+left(4x^2-25)y=0

In Problems 1–6 use (1) to find the general solution of the given differential equation on \((0, \infty)\). \(16 x^{2} y^{\prime \prime}+16 x y^{\prime}+\left(16 x^{2}-1\right) y=0\) Text Transcription: (0,infty) 16x^2y^prime\prime+16xy^prime+(16x^2-1)y=0

If n is an integer, use the substitution \(R(x)=(\alpha x)^{-1 / 2} Z(x)\) to show that the general solution of the differential equation \(x^{2} R^{\prime \prime}+2 x R^{\prime}+\left[\alpha^{2} x^{2}-n(n+1)\right] R=0\) on the interval \((0, \infty)\) is \(R(x)=c_{1} j_{n}(\alpha x)+c_{2} y_{n}(\alpha x)\), where \(j_{n}(\alpha x)\) and \(y_{n}(\alpha x)\) are the spherical Bessel functions of the first and second kind defined in (27). Text Transcription: R(x)=(alphax)^-1/2Z(x) x^2R^prime\prime+2xR^prime+[alpha^2x^2-n(n+1)] R=0 (0,infty) R(x)=c_1j_n(alphax)+c_2y_n(alphax) j_n(alphax) y_n(alphax)

In Problems 1–6 use (1) to find the general solution of the given differential equation on \((0, \infty)\). \(\frac{d}{d x}\left[x y^{\prime}\right]+\left(x-\frac{4}{x}\right) y=0\) Text Transcription: (0,infty) fracddx[xy^prime]+(x-frac4x)y=0

In Problems 7–10 use (12) to find the general solution of the given differential equation on \((0, \infty)\). \(x^{2} y^{\prime \prime}+x y^{\prime}+\left(9 x^{2}-4\right) y=0\) Text Transcription: (0,infty) x^2y^prime\prime+xy^prime+(9x^2-4)y=0

In Problems 7–10 use (12) to find the general solution of the given differential equation on \((0, \infty)\). \(x^{2} y^{\prime \prime}+x y^{\prime}+\left(36 x^{2}-\frac{1}{4}\right) y=0\) Text Transcription: (0,infty) x^2y^prime\prime+xy^prime+(36x^2-frac14)y=0

In Problems 7–10 use (12) to find the general solution of the given differential equation on \((0, \infty)\). \(x^{2} y^{\prime \prime}+x y^{\prime}+\left(25 x^{2}-\frac{4}{9}\right) y=0\) Text Transcription: (0,infty) x^2y^prime\prime+xy^prime+(25x^2-frac49)y=0

In Problems 11 and 12 use the indicated change of variable to find the general solution of the given differential equation on \((0, \infty)\). \(x^{2} y^{\prime \prime}+2 x y^{\prime}+\alpha^{2} x^{2} y=0 ; \quad y=x^{-1 / 2} v(x)\) Text Transcription: (0,infty) x^2y^prime\prime+2xy^prime+alpha^2x^2y=0;y=x^-1/2v(x)

In Problems 7–10 use (12) to find the general solution of the given differential equation on \((0, \infty)\). \(x^{2} y^{\prime \prime}+x y^{\prime}+\left(2 x^{2}-64\right) y=0\) Text Transcription: (0,infty) x^2y^prime\prime+xy^prime+(2 x^2-64)y=0

In Problems 11 and 12 use the indicated change of variable to find the general solution of the given differential equation on \((0, \infty)\). \(x^{2} y^{\prime \prime}+\left(\alpha^{2} x^{2}-\nu^{2}+\frac{1}{4}\right) y=0 ; \quad y=\sqrt{x} v(x)\) Text Transcription: (0,infty) x^2y^prime\prime+(alpha^2x^2-nu^2+frac14)y=0;y=\sqrtxv(x)

In Problems 13–20 use (18) to find the general solution of the given differential equation on \((0, \infty)\). \(x y^{\prime \prime}+2 y^{\prime}+4 y=0\) Text Transcription: (0,infty) xy^prime\prime}+2y^prime+4y=0

In Problems 13–20 use (18) to find the general solution of the given differential equation on \((0, \infty)\). \(x y^{\prime \prime}+3 y^{\prime}+x y=0\) Text Transcription: (0,infty) xy^prime\prime+3y^prime+xy=0

In Problems 13–20 use (18) to find the general solution of the given differential equation on \((0, \infty)\). \(x y^{\prime \prime}-y^{\prime}+x y=0\) Text Transcription: (0,infty) xy^prime\prime-y^prime+x y=0

In Problems 13–20 use (18) to find the general solution of the given differential equation on \((0, \infty)\). \(x y^{\prime \prime}-5 y^{\prime}+x y=0\) Text Transcription: (0,infty) xy^prime\prime-5y^prime+xy=0

In Problems 13–20 use (18) to find the general solution of the given differential equation on \((0, \infty)\). \(x^{2} y^{\prime \prime}+\left(x^{2}-2\right) y=0\) Text Transcription: (0,infty) x^2y^prime\prime+(x^2-2)y=0

In Problems 13–20 use (18) to find the general solution of the given differential equation on \((0, \infty)\). \(4 x^{2} y^{\prime \prime}+\left(16 x^{2}+1\right) y=0\) Text Transcription: (0,infty) 4x^2y^prime\prime+(16x^2+1)y=0

In Problems 13–20 use (18) to find the general solution of the given differential equation on \((0, \infty)\). \(x y^{\prime \prime}+3 y^{\prime}+x^{3} y=0\) Text Transcription: (0,infty) xy^prime\prime+3y^prime+x^{3}y=0

In Problems 13–20 use (18) to find the general solution of the given differential equation on \((0, \infty)\). \(9 x^{2} y^{\prime \prime}+9 x y^{\prime}+\left(x^{6}-36\right) y=0\) Text Transcription: (0,infty) 9x^2y^prime\prime+9xy^prime+(x^6-36)y=0

Use the series in (7) to verify that \(I_{\nu}(x)=i^{-\nu} J_{\nu}(i x)\) is a real function. Text Transcription: I_nu(x)=i^-nu}J_nu(ix)

Assume that b in equation (18) can be pure imaginary, that is, \(b=\beta i, \beta>0, i^{2}=-1\). Use this assumption to express the general solution of the given differential equation in terms of the modified Bessel functions \(I_{n}\) and \(K_{n}\). (a) \(y^{\prime \prime}-x^{2} y=0\) (b) \(x y^{\prime \prime}+y^{\prime}-7 x^{3} y=0\) Text Transcription: b=betai,beta>0,i^2=-1 I_n K_n y^prime\prime-x^2y=0 xy^prime\prime+y^prime-7x^3y=0

In Problems 23–26 first use (18) to express the general solution of the given differential equation in terms of Bessel functions. Then use (23) and (24) to express the general solution in terms of elementary functions. \(y^{\prime \prime}+y=0\) Text Transcription: y^prime\prime+y=0

In Problems 23–26 first use (18) to express the general solution of the given differential equation in terms of Bessel functions. Then use (23) and (24) to express the general solution in terms of elementary functions. \(x^{2} y^{\prime \prime}+4 x y^{\prime}+\left(x^{2}+2\right) y=0\) Text Transcription: x^2y^prime\prime+4xy^prime+(x^2+2)y=0

In Problems 23–26 first use (18) to express the general solution of the given differential equation in terms of Bessel functions. Then use (23) and (24) to express the general solution in terms of elementary functions. \(16 x^{2} y^{\prime \prime}+32 x y^{\prime}+\left(x^{4}-12\right) y=0\) Text Transcription: 16x^2y^prime\prime+32xy^prime+(x^4-12)y=0

In Problems 23–26 first use (18) to express the general solution of the given differential equation in terms of Bessel functions. Then use (23) and (24) to express the general solution in terms of elementary functions. \(4 x^{2} y^{\prime \prime}-4 x y^{\prime}+\left(16 x^{2}+3\right) y=0\) Text Transcription: 4x^2y^prime\prime-4xy^prime+(16x^2+3)y=0

(a) Proceed as in Example 5 to show that \(x J_{\nu}^{\prime}(x)=-\nu J_{\nu}(x)+x J_{\nu-1}(x)\). \([\text { Hint: Write } 2 n+\nu=2(n+\nu)-\nu .]\) (b) Use the result in part (a) to derive (21). Text Transcription: xJ_nu^prime(x)=-nuJ_nu(x)+xJ_nu-1(x) [Hint:Write2n+nu=2(n+nu)-nu]

Use the formula obtained in Example 5 along with part (a) of Problem 27 to derive the recurrence relation \(2 \nu J_{\nu}(x)=x J_{\nu+1}(x)+x J_{\nu-1}(x)\) In Problems 29 and 30 use (20) or (21) to obtain the given result. Text Transcription: 2nuJ_nu(x)=xJ_nu+1(x)+xJ_nu-1(x)

In Problems 29 and 30 use (20) or (21) to obtain the given result. \(\int_{0}^{x} r J_{0}(r) d r=x J_{1}(x)\) Text Transcription: int_0^xrJ_0(r)dr=xJ_1(x)

In Problems 29 and 30 use (20) or (21) to obtain the given result. \(J_{0}^{\prime}(x)=J_{-1}(x)=-J_{1}(x)\) Text Transcription: J_0^prime(x)=J_-1(x)=-J_1(x)

Proceed as on page 264 to derive the elementary form of \(J_{-1 / 2}(x)\) given in (24). Text Transcription: J_-1/2(x)

Use the recurrence relation in Problem 28 along with (23) and (24) to express \(J_{3 / 2}(x)\), \(J-3 / 2(x)\), \(J_{5 / 2}(x)\) and \(J_{-5 / 2}(x)\) in terms of sin x, cos x, and powers of x. Text Transcription: J_3/2(x) J-3/2(x) J_5/2(x) J_-5/2(x)

Use the change of variables \(s=\frac{2}{\alpha} \sqrt{\frac{k}{m}} e^{-\alpha t / 2}\) to show that the differential equation of the aging spring \(m x^{\prime \prime}+k e^{-\alpha t} x=0, \alpha>0\), becomes \(s^{2} \frac{d^{2} x}{d s^{2}}+s \frac{d x}{d s}+s^{2} x=0\) Text Transcription: s=frac2alphasqrtfrackme^-alphat/2 mx^prime\prime+ke^-alphatx=0,alpha>0 s^2fracd^2xds^2+sfracdxds+s^2x=0

Show that \(y=x^{1 / 2} w\left(\frac{2}{3} \alpha x^{3 / 2}\right)\) is a solution of Airy’s differential equation \(y^{\prime \prime}+\alpha^{2} x y=0, x>0\), whenever w is a solution of Bessel’s equation of order \(\frac{1}{3}\), that is \(t^{2} w^{\prime \prime}+t w^{\prime}+\left(t^{2}-\frac{1}{9}\right) w=0, t>0\). [Hint: After differentiating, substituting, and simplifying, then let \(t=\frac{2}{3} \alpha x^{3 / 2}\).] Text Transcription: y=x^1/2w(frac23alphax^3/2) y^prime\prime+alpha^2xy=0,x>0 frac13 t^2w^prime\prime+tw^prime+(t^2-frac19)w=0,t>0 t=frac23alphax^3/2

(a) Use the result of Problem 34 to express the general solution of Airy’s differential equation for x > 0 in terms of Bessel functions. (b) Verify the results in part (a) using (18).

Use the Table 6.4.1 to find the first three positive eigenvalues and corresponding eigenfunctions of the boundary-value problem \(x y^{\prime \prime}+y^{\prime}+\lambda x y=0\), \(y(x), y^{\prime}(x)\) bounded as \(x \rightarrow 0^{+}, \quad y(2)=0\) [Hint: By identifying \(\lambda=\alpha^{2}\), the DE is the parametric Bessel equation of order zero.] Text Transcription: xy^prime\prime+y^prime+lambdaxy=0 y(x),y^prime(x) xrightarrow0^+,y(2)=0 lambda=alpha^2

Use a CAS to graph \(J_{3 / 2}(x)\), \(J-3 / 2(x)\), \(J_{5 / 2}(x)\), and \(J_{-5 / 2}(x)\). Text Transcription: J_3/2(x) J-3/2(x) J_5/2(x) J_{-5/2(x)

(a) Use (18) to show that the general solution of the differential equation \(x y^{\prime \prime}+\lambda y=0\) on the interval \((0, \infty)\) is \(y=c_{1} \sqrt{x} J_{1}(2 \sqrt{\lambda x})+c_{2} \sqrt{x} Y_{1}(2 \sqrt{\lambda x})\) (b) Verify by direct substitution that \(y=\sqrt{x} J_{1}(2 \sqrt{x})\) is a particular solution of the DE in the case \(\lambda=1\). Text Transcription: xy^prime\prime+lambday=0 (0,infty) y=c_1sqrtxJ_1(2sqrtlambdax)+c_2sqrtxY_1(2sqrtlambdax) y=sqrtxJ_1(2sqrtx) lambda=1

(a) Use the general solution obtained in Problem 35 to solve the IVP \(4 x^{\prime \prime}+t x=0, \quad x(0.1)=1, \quad x^{\prime}(0.1)=-\frac{1}{2}\) Use a CAS to evaluate coefficients (b) Use a CAS to graph the solution obtained in part (a) for \(0 \leq t \leq 200\). Text Transcription: 4x^prime\prime+tx=0,x(0.1)=1,x^prime(0.1)=-frac12 0t200

Column Bending Under Its Own Weight A uniform thin column of length L, positioned vertically with one end embedded in the ground, will deflect, or bend away, from the vertical under the influence of its own weight when its length or height exceeds a certain critical value. It can be shown that the angular deflection \(\boldsymbol{\theta}\) (x) of the column from the vertical at a point P(x) is a solution of the boundary-value problem: \(E I \frac{d^{2} \theta}{d x^{2}}+\delta g(L-x) \theta=0, \quad \theta(0)=0, \quad \theta^{\prime}(L)=0\) where E is Young’s modulus, I is the cross-sectional moment of inertia \(\delta\) is the constant linear density, and x is the distance along the column measured from its base. See Figure 6.4.7. The column will bend only for those values of L for which the boundary-value problem has a nontrivial solution. (a) Restate the boundary-value problem by making the change of variables \(t=L-x\). Then use the results of a problem earlier in this exercise set to express the general solution of the differential equation in terms of Bessel functions. (b) Use the general solution found in part (a) to find a solution of the BVP and an equation which define the critical length L, that is, the smallest value of L for which the column will start to bend. (c) With the aid of a CAS, find the critical length L of a solid steel rod of radius \(r=0.05 \text { in }\), \(\delta g=0.28 \mathrm{Alb} / \text { in. }, E=2.6 \times 10^{7} \mathrm{lb} / \mathrm{in}^{2}\), \(A=\pi r^{2}\) and \(I=\frac{1}{4} \pi r^{4}\). \boldsymbol{\theta} Text Transcription: EIfracd^2thetadx^2+deltag(L-x)theta=0,theta(0)=0,theta^prime(L)=0 delta t=L-x r=0.05in deltag=0.28Albin.,E=2.6times10^7lbin^2 A=pir^2 I=frac14pir^4

Buckling of a Thin Vertical Column In Example 4 of Section 5.2 we saw that when a constant vertical compressive force, or load, P was applied to a thin column of uniform cross section and hinged at both ends, the deflection y(x) is a solution of the BVP: \(E I \frac{d^{2} y}{d x^{2}}+P y=0, \quad y(0)=0, \quad y(L)=0\) (a) If the bending stiffness factor EI is proportional to x, then \(E I(x)=k x\) where k is a constant of proportionality. If EI(L) = kL = M is the maximum stiffness factor, then k = M L and so EI(x) = Mx L. Use the information in Problem 37 to find a solution of \(M \frac{x}{L} \frac{d^{2} y}{d x^{2}}+P y=0, \quad y(0)=0, \quad y(L)=0\) if it is known that \(\sqrt{x} Y_{1}(2 \sqrt{\lambda x})\) is not zero at x 0. (b) Use Table 6.4.1 to find the Euler load \(P_{1}\) for the column. (c) Use a CAS to graph the first buckling mode \(y_{1}(x)\) corresponding to the Euler load \(P_{1}\). For simplicity assume that \(c_{1}=1\) and L = 1. Text Transcription: EIfracd^2ydx^2+Py=0,y(0)=0,y(L)=0 EI(x)=kx MfracxLfracd^2ydx^2+Py=0,y(0)=0,y(L)=0 sqrtxY_1(2sqrtlambdax) P_1 y_1(x) c_1=1

(a) Use the explicit solutions \(y_{1}(x)\) and \(y_{2}(x)\) of Legendre’s equation given in (29) and the appropriate choice of \(c_{0}\) and \(c_{1}\) to find the Legendre polynomials \(P_{6}(x)\) and \(P_{7}(x)\) (b) Write the differential equations for which \(P_{6}(x)\) and \(P_{7}(x)\) are particular solutions. Text Transcription: y_1(x) y_2(x) c_0 c_1 P_6(x) P_7

Use the recurrence relation (32) and \(P_{0}(x)=1, P_{1}(x)=x\), to generate the next six Legendre polynomials. Text Transcription: P_0(x)=1 P_1(x)=x

(a) Use the general solution given in Example 4 to solve the IVP \(4 x^{\prime \prime}+e^{-0.1 t} x=0, \quad x(0)=1, \quad x^{\prime}(0)=-\frac{1}{2}\) Also use \(J_{0}^{\prime}(x)=-J_{1}(x)\) and \(Y_{0}^{\prime}(x)=-Y_{1}(x)\) along with Table 6.4.1 or a CAS to evaluate coefficients (b) Use a CAS to graph the solution obtained in part (a) for \(0 \leq t \leq \infty\). Text Transcription: 4x^prime\prime+e^-0.1tx=0,x(0)=1,x^prime(0)=-frac12 J_0^prim}(x)=-J_1(x) Y_0^prime(x)=-Y_1(x) 0tinfty

Find the first three positive values of \(\lambda\) for which the problem \(\left(1-x^{2}\right) y^{\prime \prime}-2 x y^{\prime}+\lambda y=0\) \(y(0)=0, \quad y(x), y^{\prime}(x) \text { bounded on }[-1,1]\) has nontrivial solutions. Text Transcription: lambda (1-x^2)y^prime\prime-2 x y^prime+lambday=0 y(0)=0,y(x),y^prime(x)boundedon[-1,1]

For purposes of this problem ignore the list of Legendre polynomials given on page 266 and the graphs given in Figure 6.4.3. Use Rodrigues’ formula (33) to generate the Legendre polynomials \(P_{1}(x), P_{2}(x), \ldots, P_{7}(x)\). Use a CAS to carry out the differentiations and simplifications Text Transcription: P_1(x),P_2(x),ldots,P_7(x)

Use a root-finding application to fin the zeros of \(P_{1}(x), P_{2}(x), \ldots, P_{7}(x)\). If the Legendre polynomials are built-in functions of your CAS, fin zeros of Legendre polynomials of higher degree. Form a conjecture about the location of the zeros of any Legendre polynomial \(P_{n}(x)\), and then investigate to see whether it is true. Text Transcription: P_1(x),P_2(x),ldots,P_7(x) P_n(x)

Use a CAS to graph \(P_{1}(x), P_{2}(x), \ldots, P_{7}(x)\) on the interval \([-1,1]\). Text Transcription: P_1(x),P_2(x),ldots,P_7(x) [-1,1]

The differential equation \(y^{\prime \prime}-2 x y^{\prime}+2 \alpha y=0\) is known as Hermite’s equation of order after the French mathematician Charles Hermite (1822–1901). Show that the general solution of the equation is \(y(x)=c_{0} y_{1}(x)+c_{1} y_{2}(x)\), where \(y_{1}(x)=1+\sum_{k=1}^{\infty}(-1)^{k} \frac{2^{k} \alpha(\alpha-2) \cdots(\alpha-2 k+2)}{(2 k) !} x^{2 k}\) \(y_{2}(x)=x+\sum_{k=1}^{\infty}(-1)^{k} \frac{2^{k}(\alpha-1)(\alpha-3) \cdots(\alpha-2 k+1)}{(2 k+1) !} x^{2 k+1}\) are power series solutions centered at the ordinary point 0. Text Transcription: y^prime\prime-2xy^prime+2alphay=0 y(x)=c_0y_1(x)+c_1y_2(x) y_1(x)=1+sum_k=1^infty(-1)^kfrac2^kalpha(alpha-2)cdots(alpha-2k+2)(2 k)!x^2k y_2(x)=x+sum_k=1^infty(-1)^kfrac2^k(alpha-1)(alpha-3)cdots(alpha-2k+1)(2k+1)!x^2k+1

(a) When \(\alpha=n\) is a nonnegative integer, Hermite’s differential equation always possesses a polynomial solution of degree n. Use \(y_{1}(x)\), given in Problem 51, to find polynomial solutions for \(n=0\), \(n=2\), and \(n=4\). Then use \(y_{2}(x)\) to find polynomial solutions for \(n-1, n-3,\) and \(n-5\). (b) A Hermite polynomial \(H_{n}(x)\) is defined to be the nth degree polynomial solution of Hermite’s equation multiplied by an appropriate constant so that the coefficient of \(x^{n}\) in \(H_{n}(x)\) is \(2^{n}\). Use the polynomial solutions in part (a) to show that the first six Hermite polynomials are \begin{array}{l} H_{0}(x)=1 \\ H_{1}(x)=2 x \\ H_{2}(x)=4 x^{2}-2 \\ H_{3}(x)=8 x^{3}-12 x \\ H_{4}(x)=16 x^{4}-48 x^{2}+12 \\ H_{5}(x)=32 x^{5}-160 x^{3}+120 x\end{array} Text Transcription: alpha=n y_1(x) n=0 n=2 n=4 y_2(x) n-1 n-3 n-5 H_n(x) x^n H_n(x) 2^n H_0(x)=1 H_1(x)=2x H_2(x)=4x^2-2 H_3(x)=8x^3-12x H_4(x)=16x^4-48x^2+12 H_5(x)=32x^5-160x^3+120x

The differential equation \(\left(1-x^{2}\right) y^{\prime \prime}-x y^{\prime}+\alpha^{2} y=0\) where \(\alpha\) is a parameter, is known as Chebyshev’s equation after the Russian mathematician Pafnuty Chebyshev (1821–1894). When \(\alpha=n\) is a nonnegative integer, Chebyshev’s differential equation always possesses a polynomial solution of degree n. Find a fifth degree polynomial solution of this differential equation. Text Transcription: (1-x^2right)y^prime\prime-xy^prime+alpha^2y=0 alpha alpha=n

Pendulum of Varying Length For the simple pendulum described on page 220 of Section 5.3, suppose that the rod holding the mass m at one end is replaced by a flexible wire or string and that the wire is strung over a pulley at the point of support O in Figure 5.3.3. In this manner, while it is in motion in a vertical plane, the mass m can be raised or lowered. In other words, the length l(t) of the pendulum varies with time. Under the same assumptions leading to equation (6) in Section 5.3, it can be shown* that the differential equation for the displacement angle \(\theta\) is now \(l \theta^{\prime \prime}+2 l^{\prime} \theta^{\prime}+g \sin \theta=0\) (a) If l increases at constant rate v and if \(l(0)=l_{0}\), show that a linearization of the foregoing DE is \(\left(l_{0}+v t\right) \theta^{\prime \prime}+2 v \theta^{\prime}+g \theta=0\) (b) Make the change of variables \(x=\left(l_{0}+v t\right) v\) and show that (34) becomes \(\frac{d^{2} \theta}{d x^{2}}+\frac{2}{x} \frac{d \theta}{d x}+\frac{g}{v x} \theta=0\) (c) Use part (b) and (18) to express the general solution of equation (34) in terms of Bessel functions. (d) Use the general solution obtained in part (c) to solve the initial-value problem consisting of equation (34) and the initial conditions \(\theta(0)=\theta_{0}\), \(\theta^{\prime}(0)=0\). [Hints: To simplify calculations, use a further change of variable \(u=\frac{2}{v} \sqrt{g\left(l_{0}+v t\right)}=2 \sqrt{\frac{g}{v}} x^{1 / 2}\). Also, recall that (20) holds for both \(J_{1}(u)\) and \(Y_{1}(u)\). Finally, the identity \(J_{1}(u) Y_{2}(u)-J_{2}(u) Y_{1}(u)=-\frac{2}{\pi u}\) will be helpful.] (e) Use a CAS to graph the solution \(\theta(t)\) of the IVP in part (d) when \(l_{0}=1\) ft, \(\theta_{0}=\frac{1}{10}\) radian, and \(v=\frac{1}{60}\) ft/s. Experiment with the graph using different time intervals such as [0, 10], [0, 30], and so on. (f) What do the graphs indicate about the displacement angle \(\theta(t)\) as the length l of the wire increases with time? Text Transcription: theta ltheta^prime\prime+2l^primetheta^prime+gsintheta=0 l(0)=l_0 (l_0+vt)theta^prime\prime+2vtheta^prime+gtheta=0 x=(l_0+vt)v theta(0)=theta_0 theta^prime(0)=0 u=frac2vsqrtg(l_0+vt)=2sqrtfracgvx^1/2 J_1(u) Y_1(u) J_1(u)Y_2(u)-J_2(u)Y_1(u)=-frac2piu theta(t) l_0=1 theta_0=frac110 v=frac160

Show that the differential equation \(\sin \theta \frac{d^{2} y}{d \theta^{2}}+\cos \theta \frac{d y}{d \theta}+n(n+1)(\sin \theta) y=0\) can be transformed into Legendre’s equation by means of the substitution \(x=\cos \theta\). Text Transcription: sinthetafracd^2ydtheta^2+costhetafracdydtheta+n(n+1)(sintheta)y=0 x=cos\theta