In 15–24, x = 0 is a regular singular point of

We have seen that x = 0 is a regular singular point of any Cauchy-Euler equation \(a x^{2} y^{\prime \prime}+b x y^{\prime}+c y=0\). Are the indicial equation (14) for a Cauchy-Euler equation and its auxiliary equation related? Discuss. Text Transcription: ax^2y^prime\prime+bxy^prime+c y=0

In Problems 1–10 determine the singular points of the given differential equation. Classify each singular point as regular or irregular. \(x^{3} y^{\prime \prime}+4 x^{2} y^{\prime}+3 y=0\) Text Transcription: x^3y^prime\prime+4x^2y^prime+3y=0

In Problems 1–10 determine the singular points of the given differential equation. Classify each singular point as regular or irregular. \(x(x+3)^{2} y^{\prime \prime}-y=0\) Text Transcription: x(x+3)^2y^prime\prime-y=0

In Problems 1–10 determine the singular points of the given differential equation. Classify each singular point as regular or irregular. \(\left(x^{2}-9\right)^{2} y^{\prime \prime}+(x+3) y^{\prime}+2 y=0\) Text Transcription: (x^2-9)^2y^prime\prime+(x+3)y^prime+2y=0

In Problems 1–10 determine the singular points of the given differential equation. Classify each singular point as regular or irregular. \(y^{\prime \prime}-\frac{1}{x} y^{\prime}+\frac{1}{(x-1)^{3}} y=0\) Text Transcription: y^prime\prime-frac1xy^prime+frac1(x-1)^3y=0

In Problems 1–10 determine the singular points of the given differential equation. Classify each singular point as regular or irregular. \(\left(x^{3}+4 x\right) y^{\prime \prime}-2 x y^{\prime}+6 y=0\) Text Transcription: (x^3+4x)y^prime\prime-2xy^prime+6y=0

In Problems 1–10 determine the singular points of the given differential equation. Classify each singular point as regular or irregular. \(\left(x^{2}+x-6\right) y^{\prime \prime}+(x+3) y^{\prime}+(x-2) y=0\) Text Transcription: (x^2+x-6)y^prime\prime+(x+3)y^prime+(x-2)y=0

In Problems 1–10 determine the singular points of the given differential equation. Classify each singular point as regular or irregular. \(x^{2}(x-5)^{2} y^{\prime \prime}+4 x y^{\prime}+\left(x^{2}-25\right) y=0\) Text Transcription: x^2(x-5)^2y^prime\prime+4xy^prime+(x^2-25)y=0

In Problems 1–10 determine the singular points of the given differential equation. Classify each singular point as regular or irregular. \(x\left(x^{2}+1\right)^{2} y^{\prime \prime}+y=0\) Text Transcription: x(x^2+1)^2y^prime\prime+y=0

In Problems 1–10 determine the singular points of the given differential equation. Classify each singular point as regular or irregular. \(x^{3}\left(x^{2}-25\right)(x-2)^{2} y^{\prime \prime}+3 x(x-2) y^{\prime}+7(x+5) y=0\) Text Transcription: x^3(x^2-25)(x-2)^2y^prime\prime+3x(x-2)y^prime+7(x+5)y=0

In Problems 1–10 determine the singular points of the given differential equation. Classify each singular point as regular or irregular. \(\left(x^{3}-2 x^{2}+3 x\right)^{2} y^{\prime \prime}+x(x-3)^{2} y^{\prime}-(x+1) y=0\) Text Transcription: (x^3-2x^2+3x)^2y^prime\prime+x(x-3)^2y^prime-(x+1)y=0

In Problems 11 and 12 put the given differential equation into form (3) for each regular singular point of the equation. Identify the functions p(x) and q(x). \(x y^{\prime \prime}+(x+3) y^{\prime}+7 x^{2} y=0\) Text Transcription: x y^prime\prime+(x+3)y^prime+7x^2y=0

In Problems 13 and 14, x = 0 is a regular singular point of the given differential equation. Use the general form of the indicial equation in (14) to find the indicial roots of the singularity. Without solving, discuss the number of series solutions you would expect to find using the method of Frobenius \(x y^{\prime \prime}+y^{\prime}+10 y=0\) Text Transcription: xy^prime\prime+y^prime+10y=0

In Problems 11 and 12 put the given differential equation into form (3) for each regular singular point of the equation. Identify the functions p(x) and q(x). \(\left(x^{2}-1\right) y^{\prime \prime}+5(x+1) y^{\prime}+\left(x^{2}-x\right) y=0\) Text Transcription: (x^2-1)y^prime\prime+5(x+1)y^prime+(x^2-x)y=0

In Problems 15–24, x = 0 is a regular singular point of the given differential equation. Show that the indicial roots of the singularity do not differ by an integer. Use the method of Frobenius to obtain two linearly independent series solutions about x = 0. Form the general solution on \((0, \infty)\). \(2 x y^{\prime \prime}+5 y^{\prime}+x y=0\) Text Transcription: 2xy^prime\prime+5y^prime+xy=0

In Problems 13 and 14, x = 0 is a regular singular point of the given differential equation. Use the general form of the indicial equation in (14) to find the indicial roots of the singularity. Without solving, discuss the number of series solutions you would expect to find using the method of Frobenius \(x^{2} y^{\prime \prime}+\left(\frac{5}{3} x+x^{2}\right) y^{\prime}-\frac{1}{3} y=0\) Text Transcription: x^2y^prime\prime+(frac53x+x^2)y^prime-frac13y=0

In Problems 15–24, x = 0 is a regular singular point of the given differential equation. Show that the indicial roots of the singularity do not differ by an integer. Use the method of Frobenius to obtain two linearly independent series solutions about x = 0. Form the general solution on \((0, \infty)\). \(2 x y^{\prime \prime}-y^{\prime}+2 y=0\) Text Transcription: (0,infty) 2xy^prime\prime-y^prime+2 y=0

In Problems 15–24, x = 0 is a regular singular point of the given differential equation. Show that the indicial roots of the singularity do not differ by an integer. Use the method of Frobenius to obtain two linearly independent series solutions about x = 0. Form the general solution on \((0, \infty)\). \(4 x y^{\prime \prime}+\frac{1}{2} y^{\prime}+y=0\) Text Transcription: 4xy^prime\prime+frac12y^prime+y=0

In Problems 15–24, x = 0 is a regular singular point of the given differential equation. Show that the indicial roots of the singularity do not differ by an integer. Use the method of Frobenius to obtain two linearly independent series solutions about x = 0. Form the general solution on \((0, \infty)\). \(3 x y^{\prime \prime}+(2-x) y^{\prime}-y=0\) Text Transcription: 3xy^prime\prime+(2-x)y^prime-y=0

In Problems 15–24, x = 0 is a regular singular point of the given differential equation. Show that the indicial roots of the singularity do not differ by an integer. Use the method of Frobenius to obtain two linearly independent series solutions about x = 0. Form the general solution on \((0, \infty)\). \(2 x^{2} y^{\prime \prime}-x y^{\prime}+\left(x^{2}+1\right) y=0\) Text Transcription: 2x^2y^prime\prime-xy^prime+(x^2+1)y=0

In Problems 15–24, x = 0 is a regular singular point of the given differential equation. Show that the indicial roots of the singularity do not differ by an integer. Use the method of Frobenius to obtain two linearly independent series solutions about x = 0. Form the general solution on \((0, \infty)\). \(x^{2} y^{\prime \prime}+x y^{\prime}+\left(x^{2}-\frac{4}{9}\right) y=0\) Text Transcription: x^2y^prime\prime+xy^prime+(x^2-frac49)y=0

In Problems 15–24, x = 0 is a regular singular point of the given differential equation. Show that the indicial roots of the singularity do not differ by an integer. Use the method of Frobenius to obtain two linearly independent series solutions about x = 0. Form the general solution on \((0, \infty)\). \(x^{2} y^{\prime \prime}-\left(x-\frac{2}{9}\right) y=0\) Text Transcription: x^2y^prime\prime-(x-frac29)y=0

In Problems 15–24, x = 0 is a regular singular point of the given differential equation. Show that the indicial roots of the singularity do not differ by an integer. Use the method of Frobenius to obtain two linearly independent series solutions about x = 0. Form the general solution on \((0, \infty)\). \(9 x^{2} y^{\prime \prime}+9 x^{2} y^{\prime}+2 y=0\) Text Transcription: 9x^2y^prime\prime+9x^2y^prime+2y=0

In Problems 15–24, x = 0 is a regular singular point of the given differential equation. Show that the indicial roots of the singularity do not differ by an integer. Use the method of Frobenius to obtain two linearly independent series solutions about x = 0. Form the general solution on \((0, \infty)\). \(2 x y^{\prime \prime}-(3+2 x) y^{\prime}+y=0\) Text Transcription: 2xy^prime\prime-(3+2 x)y^prime+y=0

In Problems 15–24, x = 0 is a regular singular point of the given differential equation. Show that the indicial roots of the singularity do not differ by an integer. Use the method of Frobenius to obtain two linearly independent series solutions about x = 0. Form the general solution on \((0, \infty)\). \(2 x^{2} y^{\prime \prime}+3 x y^{\prime}+(2 x-1) y=0\) Text Transcription: 2x^2y^prime\prime+3xy^prime+(2x-1)y=0

In Problems 25–30, x = 0 is a regular singular point of the given differential equation. Show that the indicial roots of the singularity differ by an integer. Use the method of Frobenius to obtain at least one series solution about x 0. Use (23) where necessary and a CAS, if instructed, to find a second solution. Form the general solution on \((0, \infty)\). \(x y^{\prime \prime}+2 y^{\prime}-x y=0\) Text Transcription: (0,infty) x y^prime\prime+2y^prime-xy=0

In Problems 25–30, x = 0 is a regular singular point of the given differential equation. Show that the indicial roots of the singularity differ by an integer. Use the method of Frobenius to obtain at least one series solution about x 0. Use (23) where necessary and a CAS, if instructed, to find a second solution. Form the general solution on \((0, \infty)\). \(x^{2} y^{\prime \prime}+x y^{\prime}+\left(x^{2}-\frac{1}{4}\right) y=0\) Text Transcription: (0,infty) x^2y^prime\prime+xy^prime+left(x^2-frac14)y=0

In Problems 25–30, x = 0 is a regular singular point of the given differential equation. Show that the indicial roots of the singularity differ by an integer. Use the method of Frobenius to obtain at least one series solution about x 0. Use (23) where necessary and a CAS, if instructed, to find a second solution. Form the general solution on \((0, \infty)\). \(x y^{\prime \prime}+y^{\prime}+y=0\) Text Transcription: (0,infty) xy^prime\prime+y^prime+y=0

In Problems 25–30, x = 0 is a regular singular point of the given differential equation. Show that the indicial roots of the singularity differ by an integer. Use the method of Frobenius to obtain at least one series solution about x 0. Use (23) where necessary and a CAS, if instructed, to find a second solution. Form the general solution on \((0, \infty)\). \(x y^{\prime \prime}-x y^{\prime}+y=0\) Text Transcription: (0,infty) xy^prime\prime-xy^prime+y=0

In Problems 25–30, x = 0 is a regular singular point of the given differential equation. Show that the indicial roots of the singularity differ by an integer. Use the method of Frobenius to obtain at least one series solution about x 0. Use (23) where necessary and a CAS, if instructed, to find a second solution. Form the general solution on \((0, \infty)\). \(y^{\prime \prime}+\frac{3}{x} y^{\prime}-2 y=0\) Text Transcription: (0,infty) y^prime\prime+frac3xy^prime-2 y=0

In Problems 31 and 32, x = 0 is a regular singular point of the given differential equation. Show that the indicial roots of the singularity differ by an integer. Use the recurrence relation found by the method of Frobenius first with the larger root \(r_{1}\). How many solutions did you find? Next use the recurrence relation with the smaller root \(r_{2}\). How many solutions did you find? \(x y^{\prime \prime}+(x-6) y^{\prime}-3 y=0\) Text Transcription: r_1 r_2 xy^prime\prime+(x-6)y^prime-3y=0

In Problems 25–30, x = 0 is a regular singular point of the given differential equation. Show that the indicial roots of the singularity differ by an integer. Use the method of Frobenius to obtain at least one series solution about x 0. Use (23) where necessary and a CAS, if instructed, to find a second solution. Form the general solution on \((0, \infty)\). \(x y^{\prime \prime}+(1-x) y^{\prime}-y=0\) Text Transcription: (0,infty) xy^prime\prime+(1-x)y^prime-y=0

In Problems 31 and 32, x = 0 is a regular singular point of the given differential equation. Show that the indicial roots of the singularity differ by an integer. Use the recurrence relation found by the method of Frobenius first with the larger root \(r_{1}\). How many solutions did you find? Next use the recurrence relation with the smaller root \(r_{2}\). How many solutions did you find? \(x(x-1) y^{\prime \prime}+3 y^{\prime}-2 y=0\) Text Transcription: r_1 r_2 x(x-1)y^prime\prime+3y^prime-2y=0

(a) The differential equation \(x^{4} y^{\prime \prime}+\lambda y=0\) has an irregular singular point at x = 0. Show that the substitution t = 1 x yields the DE \(\frac{d^{2} y}{d t^{2}}+\frac{2}{t} \frac{d y}{d t}+\lambda y=0\) which now has a regular singular point at t = 0. (b) Use the method of this section to find two series solutions of the second equation in part (a) about the regular singular point t = 0. (c) Express each series solution of the original equation in terms of elementary functions. Text Transcription: x^4y^prime\prime}+lambday=0 fracd^2yd t^2+frac2tfracdydt+lambday=0

Each of the differential equations \(x^{3} y^{\prime \prime}+y=0\) and \(x^{2} y^{\prime \prime}+(3 x-1) y^{\prime}+y=0\) has an irregular singular point at x = 0. Determine whether the method of Frobenius yields a series solution of each differential equation about x = 0. Discuss and explain your findings Text Transcription: x^3y^prime\prime+y=0 x^2y^prime\prime+(3x-1)y^prime+y=0\)

Buckling of a Tapered 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, the deflection y(x) was a solution of the boundary-value problem \(E I \frac{d^{2} y}{d x^{2}}+P y=0, \quad y(0)=0, \quad y(L)=0\) The assumption here is that the column is hinged at both ends. The column will buckle or deflect only when the compressive force is a critical load \(P_{n}\). (a) In this problem let us assume that the column is of length L, is hinged at both ends, has circular cross sections, and is tapered as shown in Figure 6.3.1(a). If the column, a truncated cone, has a linear taper y cx as shown in cross section in Figure 6.3.1(b), the moment of inertia of a cross section with respect to an axis perpendicular to the xy-plane is \(I=\frac{1}{4} \pi r^{4}\), where \(r=y\) and y = cx. Hence we can write \(I(x)=I_{0}(x b)^{4}\), where \(I_{0}=I(b)=\frac{1}{4} \pi(c b)^{4}\) Substituting I(x) into the differential equation in (24), we see that the deflection in this case is determined from the BVP \(x^{4} \frac{d^{2} y}{d x^{2}}+\lambda y=0, \quad y(a)=0, \quad y(b)=0\), where \(\lambda=P b^{4} \quad E I_{0}\). Use the results of Problem 33 to find the critical loads \(P_{n}\) for the tapered column. Use an appropriate identity to express the buckling modes yn(x) as a single function. (b) Use a CAS to plot the graph of the first buckling mode \(y_{1}(x)\) corresponding to the Euler load \(P_{1}\) when b = 11 and a = 1. Text Transcription: EIfracd^2ydx^2+Py=0,y(0)=0,y(L)=0 P_n I=frac14pir^4 r=y I(x)=I_0(xb)^4 I_0=I(b)=frac14pi(cb)^4 y_1(x) P_1

Discuss how you would define a regular singular point for the linear third-order differential equation \(a_{3}(x) y^{\prime \prime \prime}+a_{2}(x) y^{\prime \prime}+a_{1}(x) y^{\prime}+a_{0}(x) y=0\) Text Transcription: a_3(x)y^prime\prime\prime+a_2(x)y^prime\prime+a_1(x)y^prime+a_0(x)y=0