Chapter 3

The given differential equation is a model of an undamped spring/mass system in which the restoring force \(F(x)\) in (1) is nonlinear. For each equation use a numerical solver to plot the solution curves satisfying the given initial conditions. If the solutions appear to be periodic, use the solution curve to estimate the period \(T\) of oscillations. $$ \begin{aligned} &\frac{d^{2} x}{d t^{2}}+x^{3}=0 \\ &x(0)=1, x^{\prime}(0)=1 ; x(0)=\frac{1}{2}, x^{\prime}(0)=-1 \end{aligned} $$

Solve the given system of differential equations by systematic elimination. $$ \begin{aligned} &\frac{d x}{d t}=2 x-y \\ &\frac{d y}{d t}=x \end{aligned} $$

In Problems 1-26, solve the given differential equation by undetermined coefficients. \(y^{\prime \prime}+3 y^{\prime}+2 y=6\)

In Problems, 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.. $$ \left(y^{\prime \prime}\right)^{2}=y^{2} ; y_{1}=e^{x}, y_{2}=\cos x $$

A mass weighing 4 pounds is attached to a spring whosespring constant is \(16 \mathrm{lb} / \mathrm{ft}\). What is the period of simple harmonic motion?

Solve the given differential equation. $$ x^{2} y^{\prime \prime}-2 y=0 $$

The given family of functions is the general solution of the differential equation on the indicated interval. Find a member of the family that is a solution of the initial-value problem. $$ y=c_{1} e^{x}+c_{2} e^{-x},(-\infty, \infty) ; y^{\prime \prime}-y=0, y(0)=0, y^{\prime}(0)=1 $$

In Problems \(1-20\), solve the given system of differential equations by systematic elimination. $$ \begin{aligned} &\frac{d x}{d t}=2 x-y \\ &\frac{d y}{d t}=x \end{aligned} $$

In Problems \(1-4\), the given differential equation is a model of an undamped spring/mass system in which the restoring force \(F(x)\) in (1) is nonlinear. For each equation use a numerical solver to plot the solution curves satisfying the given initial conditions. If the solutions appear to be periodic, use the solution curve to estimate the period \(T\) of oscillations. $$ \frac{d^{2} x}{d t^{2}}+x^{3}=0 $$ $$ x(0)=1, x^{\prime}(0)=1 ; x(0)=\frac{1}{2}, x^{\prime}(0)=-1 $$

In Problems 1-18, solve the given differential equation. $$ x^{2} y^{\prime \prime}-2 y=0 $$

In Problems \(1-18\), solve each differential equation by variation of parameters. $$ y^{\prime \prime}+y=\sec x $$

In Problems \(1-16\), the indicated function \(y_{1}(x)\) is a solution of the given equation. Use reduction of order or formula (5), as instructed, to find a second solution \(y_{2}(x)\). $$ y^{\prime \prime}-4 y^{\prime}+4 y=0 ; \quad y_{1}=e^{2 x} $$

In Problems \(1-4\), the given family of functions is the general solution of the differential equation on the indicated interval. Find a member of the family that is a solution of the initial-value problem. $$ y=c_{1} e^{x}+c_{2} e^{-x},(-\infty, \infty) ; y^{\prime \prime}-y=0, y(0)=0, y^{\prime}(0)=1 $$

The given differential equation is a model of an undamped spring/mass system in which the restoring force \(F(x)\) in (1) is nonlinear. For each equation use a numerical solver to plot the solution curves satisfying the given initial conditions. If the solutions appear to be periodic, use the solution curve to estimate the period \(T\) of oscillations. $$ \begin{aligned} &\frac{d^{2} x}{d t^{2}}+4 x-16 x^{3}=0 \\ &x(0)=1, x^{\prime}(0)=1 ; x(0)=-2, x^{\prime}(0)=2 \end{aligned} $$

Solve the given system of differential equations by systematic elimination. $$ \begin{aligned} &\frac{d x}{d t}=4 x+7 y \\ &\frac{d y}{d t}=x-2 y \end{aligned} $$

Solve the given differential equation by undetermined coefficients. \(4 y^{\prime \prime}+9 y=15\)

In Problems, 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. $$ y y^{\prime \prime}=\frac{1}{2}\left(y^{\prime}\right)^{2} ; y_{1}=1, y_{2}=x^{2} $$

A 20 -kilogram mass is attached to a spring. If the frequency of simple harmonic motion is \(2 / \pi\) cycles/s, what is the spring constant \(k ?\) What is the frequency of simple harmonic motion if the original mass is replaced with an 80 -kilogram mass?

Solve the given differential equation. $$ 4 x^{2} y^{\prime \prime}+y=0 $$

The given family of functions is the general solution of the differential equation on the indicated interval. Find a member of the family that is a solution of the initial-value problem. $$ \begin{aligned} &y=c_{1} e^{4 x}+c_{2} e^{-x},(-\infty, \infty) ; y^{\prime \prime}-3 y^{\prime}-4 y=0, y(0)=1, \\ &y^{\prime}(0)=2 \end{aligned} $$

Answer Problems \(1-8\) without referring back to the text. Fill in the blank or answer true/false. For the method of undetermined coefficients, the assumed form of the particular solution \(y_{p}\) for \(y^{\prime \prime}-y=1+e^{x}\) is _________.

In Problems \(1-20\), solve the given system of differential equations by systematic elimination. $$ \begin{aligned} &\frac{d x}{d t}=4 x+7 y \\ &\frac{d y}{d t}=x-2 y \end{aligned} $$

In Problems \(1-4\), the given differential equation is a model of an undamped spring/mass system in which the restoring force \(F(x)\) in (1) is nonlinear. For each equation use a numerical solver to plot the solution curves satisfying the given initial conditions. If the solutions appear to be periodic, use the solution curve to estimate the period \(T\) of oscillations. $$ \begin{aligned} &d^{2} x \\ &d t^{2}+4 x-16 x^{3}=0 \\ &x(0)=1, x^{\prime}(0)=1 ; x(0)=-2, x^{\prime}(0)=2 \end{aligned} $$

In Problems 1-18, solve the given differential equation. $$ 4 x^{2} y^{\prime \prime}+y=0 $$

In Problems \(1-18\), solve each differential equation by variation of parameters. $$ y^{\prime \prime}+y=\tan x $$

In Problems 1-26, solve the given differential equation by undetermined coefficients. $$ 4 y^{\prime \prime}+9 y=15 $$

In Problems \(1-16\), the indicated function \(y_{1}(x)\) is a solution of the given equation. Use reduction of order or formula (5), as instructed, to find a second solution \(y_{2}(x)\). $$ y^{\prime \prime}+2 y^{\prime}+y=0 ; \quad y_{1}=x e^{-x} $$

In Problems \(1-4\), the given family of functions is the general solution of the differential equation on the indicated interval. Find a member of the family that is a solution of the initial-value problem. $$ \begin{aligned} &y=c_{1} e^{4 x}+c_{2} e^{-x},(-\infty, \infty) ; y^{\prime \prime}-3 y^{\prime}-4 y=0, y(0)=1 \\ &y^{\prime}(0)=2 \end{aligned} $$

The given differential equation is a model of an undamped spring/mass system in which the restoring force \(F(x)\) in (1) is nonlinear. For each equation use a numerical solver to plot the solution curves satisfying the given initial conditions. If the solutions appear to be periodic, use the solution curve to estimate the period \(T\) of oscillations. $$ \begin{aligned} &\frac{d^{2} x}{d t^{2}}+2 x-x^{2}=0 \\ &x(0)=1, x^{\prime}(0)=1 ; x(0)=\frac{3}{2}, x^{\prime}(0)=-1 \end{aligned} $$

Solve the given system of differential equations by systematic elimination. $$ \begin{aligned} &\frac{d x}{d t}=-y+t \\ &\frac{d y}{d t}=x-t \end{aligned} $$

Solve the given differential equation by undetermined coefficients. \(y^{\prime \prime}-10 y^{\prime}+25 y=30 x+3\)

In Problems, solve the given differential equation by using the substitution \(u=y^{\prime}\). $$ y^{\prime \prime}+\left(y^{\prime}\right)^{2}+1=0 $$

Solve the given differential equation. $$ x y^{\prime \prime}+y^{\prime}=0 $$

Solve each differential equation by variation of parameters. $$ y^{\prime \prime}+y=\sin x $$

The given family of functions is the general solution of the differential equation on the indicated interval. Find a member of the family that is a solution of the initial-value problem. $$ \begin{aligned} &y=c_{1} x+c_{2} x \ln x,(0, \infty) ; x^{2} y^{\prime \prime}-x y^{\prime}+y=0, y(1)=3 \\ &y^{\prime}(1)=-1 \end{aligned} $$

Answer Problems \(1-8\) without referring back to the text. Fill in the blank or answer true/false. A constant multiple of a solution of a linear differential equation is also a solution.

In Problems \(1-20\), solve the given system of differential equations by systematic elimination. $$ \begin{aligned} &\frac{d x}{d t}=-y+t \\ &\frac{d y}{d t}=x-t \end{aligned} $$

In Problems \(1-4\), the given differential equation is a model of an undamped spring/mass system in which the restoring force \(F(x)\) in (1) is nonlinear. For each equation use a numerical solver to plot the solution curves satisfying the given initial conditions. If the solutions appear to be periodic, use the solution curve to estimate the period \(T\) of oscillations. $$ \begin{aligned} &\frac{d^{2} x}{d t^{2}}+2 x-x^{2}=0 \\ &x(0)=1, x^{\prime}(0)=1 ; x(0)=\frac{3}{2}, x^{\prime}(0)=-1 \end{aligned} $$

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

In Problems 1-18, solve the given differential equation. $$ x y^{\prime \prime}+y^{\prime}=0 $$

In Problems \(1-18\), solve each differential equation by variation of parameters. $$ y^{\prime \prime}+y=\sin x $$

In Problems 1-26, solve the given differential equation by undetermined coefficients. $$ y^{\prime \prime}-10 y^{\prime}+25 y=30 x+3 $$

In Problems \(1-16\), the indicated function \(y_{1}(x)\) is a solution of the given equation. Use reduction of order or formula (5), as instructed, to find a second solution \(y_{2}(x)\). $$ y^{\prime \prime}+16 y=0 ; \quad y_{1}=\cos 4 x $$

In Problems \(1-4\), the given family of functions is the general solution of the differential equation on the indicated interval. Find a member of the family that is a solution of the initial-value problem. $$ \begin{aligned} &y=c_{1} x+c_{2} x \ln x,(0, \infty) ; x^{2} y^{\prime \prime}-x y^{\prime}+y=0, y(1)=3 \\ &y^{\prime}(1)=-1 \end{aligned} $$

The given differential equation is a model of an undamped spring/mass system in which the restoring force \(F(x)\) in (1) is nonlinear. For each equation use a numerical solver to plot the solution curves satisfying the given initial conditions. If the solutions appear to be periodic, use the solution curve to estimate the period \(T\) of oscillations. $$ \begin{aligned} &\frac{d^{2} x}{d t^{2}}+x e^{0.01 x}=0 \\ &x(0)=1, x^{\prime}(0)=1 ; x(0)=3, x^{\prime}(0)=-1 \end{aligned} $$

Solve the given system of differential equations by systematic elimination. $$ \begin{aligned} &\frac{d x}{d t}-4 y=1 \\ &\frac{d y}{d t}+x=2 \end{aligned} $$

Solve the given differential equation by undetermined coefficients. \(y^{\prime \prime}+y^{\prime}-6 y=2 x\)

In Problems, solve the given differential equation by using the substitution \(u=y^{\prime}\). $$ y^{\prime \prime}=1+\left(y^{\prime}\right)^{2} $$

Solve the given differential equation. $$ x y^{\prime \prime}-3 y^{\prime}=0 $$

Find the general solution of the given second-order differential equation. $$ y^{\prime \prime}-3 y^{\prime}+2 y \quad 0 $$