Solved: In 1 and 2 verify that y1 and y2 are solutions of

In Problems 3–8 solve the given differential equation by using the substitution \(u=y^{\prime}\). \(y^{\prime \prime}+\left(y^{\prime}\right)^{2}+1=0\) Text Transcription: u=y^{\prime} y^{\prime \prime}+\left(y^{\prime}\right)^{2}+1=0

In Problems 1 and 2 verify that \(y_{1}\) and \(y_{2}\) are solutions of the given differential equation but that \(y=c_{1} y_{1}+c_{2} y_{2}\) is, in general, not a solution. \(y y^{\prime \prime}=\frac{1}{2}\left(y^{\prime}\right)^{2} ; \quad y_{1}=1, y_{2}=x^{2}\) Text Transcription: y_1 y_2 y=c_1y_1+c_2y_2 yy^prime=frac12y^primeright^2;y_1=1y_2=x^2

In Problems 3–8 solve the given differential equation by using the substitution \(u=y^{\prime}\). \(y^{\prime \prime}=1+\left(y^{\prime}\right)^{2}\) Text Transcription: u=y^{\prime} y^prime\prime=1+y^prime^2

In Problems 3–8 solve the given differential equation by using the substitution \(u=y^{\prime}\). \(x^{2} y^{\prime \prime}+\left(y^{\prime}\right)^{2}=0\) Text Transcription: u=y^{\prime} x^2y^prime+y^prime^2=0

A mathematical model for the position x(t) of a moving object is \(\frac{d^{2} x}{d t^{2}}+\sin x=0\) Use a numerical solver to graphically investigate the solutions of the equation subject to \(x(0)=0, x^{\prime}(0)=x_{1}\) \(x_{1} \geq 0\). Discuss the motion of the object for \(t \geq 0\) and for various choices of x1. Investigate the equation \(\frac{d^{2} x}{d t^{2}}+\frac{d x}{d t}+\sin x=0\) in the same manner. Give a possible physical interpretation of the dxdt term. Text Transcription: d^2xdt^2+sinx=0 x(0)=0,x^prime(0)=x_1 tgeq 0 d^2xdt^2+dxdt+sinx=0

In Problems 3–8 solve the given differential equation by using the substitution \(u=y^{\prime}\). \((y+1) y^{\prime \prime}=\left(y^{\prime}\right)^{2}\) Text Transcription: u=y^{\prime} y+1y^prime\prime=y^prime^2

In Problems 3–8 solve the given differential equation by using the substitution \(u=y^{\prime}\). \(y^{\prime \prime}+2 y\left(y^{\prime}\right)^{3}=0\) Text Transcription: u=y^{\prime} y^prime\prime+2yy^prime^3=0

In Problems 3–8 solve the given differential equation by using the substitution \(u=y^{\prime}\). \(y^{2} y^{\prime \prime}=y^{\prime}\) Text Transcription: u=y^{\prime} y^2y^prime\prime=y^prime

Consider the initial-value problem. \(y^{\prime \prime}+y y^{\prime}=0, \quad y(0)=1, y^{\prime}(0)=-1\) (a) Use the DE and a numerical solver to graph the solution curve. (b) Find an explicit solution of the IVP. Use a graphing utility to graph this solution. (c) Find an interval of definition for the solution in part (b). Text Transcription: y^prime\prime+yy^prime=0y(0)=1,y^prime(0)=-1

Find two solutions of the initial-value problem \(\left(y^{\prime \prime}\right)^{2}+\left(y^{\prime}\right)^{2}=1, \quad y\left(\frac{\pi}{2}\right)=\frac{1}{2}, \quad y^{\prime}\left(\frac{\pi}{2}\right)=\frac{\sqrt{3}}{2} .\) Use a numerical solver to graph the solution curves. Text Transcription: y^prime\prime^2+y^prime^2=1,ypi2=frac12,y^primefracpi2fracsqrt32.

In Problems 13 and 14 show that the substitution \(u=y^{\prime}\) leads to a Bernoulli equation. Solve this equation (see Section 2.5). \(x y^{\prime \prime}=y^{\prime}+\left(y^{\prime}\right)^{3}\) Text Transcription: u=y^prime xy^prime\prime=y^prime+y^prime^3

In Problems 13 and 14 show that the substitution \(u=y^{\prime}\) leads to a Bernoulli equation. Solve this equation (see Section 2.5). \(x y^{\prime \prime}=y^{\prime}+x\left(y^{\prime}\right)^{2}\) Text Transcription: u=y^prime x y^prime\prime=y^prime+xy^prime^2

In Problems 15–18 proceed as in Example 3 and obtain the first six nonzero terms of a Taylor series solution, centered at 0, of the given initial-value problem. Use a numerical solver and a graphing utility to compare the solution curve with the graph of the Taylor polynomial. \(y^{\prime \prime}=x^{2}+y^{2}-2 y^{\prime}, \quad y(0)=1, y^{\prime}(0)=1\) Text Transcription: y^prime\prime=x^2+y^2-2y^prime,y(0)=1,y^prime(0)=1

In Problems 9 and 10 solve the given initial-value problem. \(2 y^{\prime} y^{\prime \prime}=1, y(0)=2, y^{\prime}(0)=1\) Text Transcription: 2 y^{\prime} y^{\prime \prime}=1, y(0)=2, y^{\prime}(0)=1

In Problems 9 and 10 solve the given initial-value problem. \(y^{\prime \prime}+x\left(y^{\prime}\right)^{2}=0, y(1)=4, y^{\prime}(1)=2\) Text Transcription: y^prime\prime+xy^prime^2=0,y(1)=4,y^prime(1)=2

In Problems 15–18 proceed as in Example 3 and obtain the first six nonzero terms of a Taylor series solution, centered at 0, of the given initial-value problem. Use a numerical solver and a graphing utility to compare the solution curve with the graph of the Taylor polynomial. \(y^{\prime\prime}=e^y,\ y(0)=0,\ y^\prime(0)=-1\) Text Transcription: y^{\prime \prime}=e^{y}, \quad y(0)=0, y^{\prime}(0)=-1

In calculus the curvature of a curve that is defined by a function \(y=f(x)\) is defined a \(\kappa=\frac{y^{\prime \prime}}{\left[1+\left(y^{\prime}\right)^{2}\right]^{3 / 2}}\) Find \(y=f(x)\) for which \(\boldsymbol{\kappa}=1\). [Hint: For simplicity, ignore constants of integration.] Text Transcription: y=f(x) \kappa=\frac{y^{\prime \prime}}{\left[1+\left(y^{\prime}\right)^{2}\right]^{3 / 2}}

Discuss how to find an alternative two-parameter family of solutions for the nonlinear differential equation \(y^{\prime \prime}=2 x\left(y^{\prime}\right)^{2}\) in Example 1. [Hint: Suppose that \(-c_{1}^{2}\) is used as the constant of integration instead of \(+c_{1}^{2}\).] Text Transcription: y^prime\prime=2xy^prime^2 -c_1^2

In Problem 1 we saw that cos x and \(e^{x}\) were solutions of the nonlinear equation \(\left(y^{\prime \prime}\right)^{2}-y^{2}=0\). Verify that sin x and \(e^{-x}\) are also solutions. Without attempting to solve the differential equation, discuss how these explicit solutions can be found by using knowledge about linear equations. Without attempting to verify, discuss why the linear combinations \(y=c_{1} e^{x}+c_{2} e^{-x}+c_{3} \cos x+c_{4} \sin x\) and \(y=c_{2} e^{-x}+c_{4} \sin x\) are not, in general, solutions, but the two special linear combinations \(y=c_{1} e^{x}+c_{2} e^{-x}\) and \(y=c_{3} \cos x+c_{4} \sin x\) must satisfy the differential equation. Text Transcription: e^x y^prime\prime^2-y^2=0 e^-x y=c_1e^x+c_2e^-x+c_3cosx+c_4sin x y=c_1e^x+c_2e^-x} y=c_3cosx+c_4sin x

Motion in a Force Field A mathematical model for the position x(t) of a body moving rectilinearly on the x-axis in an inverse-square force field is given b \(\frac{d^{2} x}{d t^{2}}=-\frac{k^{2}}{x^{2}}\) Suppose that at \(t=0\) the body starts from rest from the position \(x=x_{0}, \quad x_{0}>0\). Show that the velocity of the body at time t is given by \(v^{2}=2 k^{2}\left(1 / x-1 / x_{0}\right)\). Use the last expression and a CAS to carry out the integration to express time t in terms of x. Text Transcription: fracd^2xdt^2=-frack^2x^2 t=0 x=x_0,x_0>0 v^2=2k^21/x-1/x_0

Discuss how the method of reduction of order considered in this section can be applied to the third-order differential equation \(y^{\prime \prime \prime}=\sqrt{1+\left(y^{\prime \prime}\right)^{2}}\). Carry out your ideas and solve the equation. Text Transcription: y^prime\prime\prime=sqrt1+y^prime\prime^2

In Problems 1 and 2 verify that \(y_{1}\) and \(y_{2}\) are solutions of the given differential equation but that \(y=c_{1} y_{1}+c_{2} y_{2}\) is, in general, not a solution. \(\left(y^{\prime \prime}\right)^{2}=y^{2} ; \quad y_{1}=e^{x}, y_{2}=\cos x\) Text Transcription: y_1 y_2 y=c_1y_1+c_2y_2 y^prime^2=y^2};y_1=e^x,y_2=cosx

In Problems 15–18 proceed as in Example 3 and obtain the first six nonzero terms of a Taylor series solution, centered at 0, of the given initial-value problem. Use a numerical solver and a graphing utility to compare the solution curve with the graph of the Taylor polynomial. \(y^{\prime \prime}=x+y^{2}, \quad y(0)=1, y^{\prime}(0)=1\) Text Transcription: y^prime\prime=x+y^2,\quady(0)=1,y^prime(0)=1

In Problems 15–18 proceed as in Example 3 and obtain the first six nonzero terms of a Taylor series solution, centered at 0, of the given initial-value problem. Use a numerical solver and a graphing utility to compare the solution curve with the graph of the Taylor polynomial. \(y^{\prime \prime}+y^{2}=1, \quad y(0)=2, y^{\prime}(0)=3\) Text Transcription: y^prime\prime+y^2=1,y(0)=2, y^prime(0)=3