Chapter 3

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. $$ \begin{aligned} &y=c_{1}+c_{2} \cos x+c_{3} \sin x,(-\infty, \infty) ; y^{\prime \prime \prime}+y^{\prime}=0 \\ &y(\pi)=0, y^{\prime}(\pi)=2, y^{\prime \prime}(\pi)=-1 \end{aligned} $$

Answer Problems \(1-8\) without referring back to the text. Fill in the blank or answer true/false. If \(f_{1}\) and \(f_{2}\) are linearly independent functions on an interval \(I\), then their Wronskian \(W\left(f_{1}, f_{2}\right) \neq 0\) for all \(x\) in \(I\).

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

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. $$ \begin{aligned} &\frac{d^{2} x}{d t^{2}}+x e^{0.01 x}=0 \\ &x(0)=1, x^{\prime}(0)=1 ; x(0)=3, x^{\prime}(0)=-1 \end{aligned} $$

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

In Problems 1-18, solve the given differential equation. $$ x y^{\prime \prime}-3 y^{\prime}=0 $$

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

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 \(y_{2}(x)\). $$ y^{\prime \prime}+9 y=0 ; \quad y_{1}=\sin 3 x $$

Solve the given system of differential equations by systematic elimination. $$ \begin{aligned} &\left(D^{2}+5\right) x-2 y=0 \\ &-2 x+\left(D^{2}+2\right) y=0 \end{aligned} $$

Solve the given differential equation by undetermined coefficients. \(\frac{1}{4} y^{\prime \prime}+y^{\prime}+y=x^{2}-2 x\)

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

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

Given that \(y=c_{1}+c_{2} x^{2}\) is a two-parameter family of solutions of \(x y^{\prime \prime}-y^{\prime}=0\) on the interval \((-\infty, \infty)\), show that constants \(c_{1}\) and \(c_{2}\) cannot be found so that a member of the family satisfies the initial conditions \(y(0)=0, y^{\prime}(0)=1\). Explain why this does not violate Theorem 3.1.1.

In Problems \(1-20\), solve the given system of differential equations by systematic elimination. $$ \begin{aligned} &\left(D^{2}+5\right) x-2 y=0 \\ &-2 x+\left(D^{2}+2\right) y=0 \end{aligned} $$

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

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

In Problems \(1-18\), solve each differential equation by variation of parameters. $$ y^{\prime \prime}+y=\cos ^{2} x $$

In Problems 1-26, solve the given differential equation by undetermined coefficients. $$ \frac{1}{4} y^{\prime \prime}+y^{\prime}+y=x^{2}-2 x $$

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 \(y_{2}(x)\). $$ y^{\prime \prime}-y=0 ; \quad y_{1}=\cosh x $$

Solve the given system of differential equations by systematic elimination. $$ \begin{aligned} (D+1) x+(D-1) y &=2 \\ 3 x+(D+2) y &=-1 \end{aligned} $$

Solve the given differential equation by undetermined coefficients. \(y^{\prime \prime}-8 y^{\prime}+20 y=100 x^{2}-26 x e^{x}\)

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

A force of 400 newtons stretches a spring 2 meters. \(\mathrm{A}\) mass of 50 kilograms is attached to the end of the spring and is initially released from the equilibrium position with an upward velocity of \(10 \mathrm{~m} / \mathrm{s}\). Find the equation of motion.

Solve the given differential equation. $$ x^{2} y^{\prime \prime}+5 x y^{\prime}+3 y=0 $$

Find the general solution of the given second-order differential equation. $$ y^{\prime \prime}-10 y^{\prime}+25 y \quad 0 $$

Solve each differential equation by variation of parameters. $$ y^{\prime \prime}+y=\sec ^{2} x $$

In Problems \(1-20\), solve the given system of differential equations by systematic elimination. $$ \begin{aligned} (D+1) x+(D-1) y &=2 \\ 3 x+(D+2) y &=-1 \end{aligned} $$

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

In Problems 1-18, solve the given differential equation. $$ x^{2} y^{\prime \prime}+5 x y^{\prime}+3 y=0 $$

In Problems 1-26, solve the given differential equation by undetermined coefficients. $$ y^{\prime \prime}-8 y^{\prime}+20 y=100 x^{2}-26 x e^{x} $$

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 \(y_{2}(x)\). $$ y^{\prime \prime}-25 y=0 ; \quad y_{1}=e^{5 x} $$

Solve the given differential equation by undetermined coefficients. \(y^{\prime \prime}+3 y=-48 x^{2} e^{3 x}\)

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

Solve the given differential equation. $$ x^{2} y^{\prime \prime}-3 x y^{\prime}-2 y=0 $$

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

Given that \(x(t)=c_{1} \cos \omega t+c_{2} \sin \omega t\) is the general solution of \(x^{\prime \prime}+\omega^{2} x=0\) on the interval \((-\infty, \infty)\), show that a solution satisfying the initial conditions \(x(0)=x_{0}, x^{\prime}(0)=x_{1}\), is given by $$ x(t)=x_{0} \cos \omega t+\frac{x_{1}}{\omega} \sin \omega t . $$

Answer Problems \(1-8\) without referring back to the text. Fill in the blank or answer true/false. The differential equation describing the motion of a mass attached to a spring is \(x^{\prime \prime}+16 x=0\). If the mass is released at \(t=0\) from 1 meter above the equilibrium position with a downward velocity of \(3 \mathrm{~m} / \mathrm{s}\), the amplitude of vibrations is _________ meters.

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

In Problems 1-18, solve the given differential equation. $$ x^{2} y^{\prime \prime}-3 x y^{\prime}-2 y=0 $$

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

In Problems 1-26, solve the given differential equation by undetermined coefficients. $$ y^{\prime \prime}+3 y=-48 x^{2} e^{3 x} $$

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 \(y_{2}(x)\). $$ 9 y^{\prime \prime}-12 y^{\prime}+4 y=0 ; \quad y_{1}=e^{2 x / 3} $$

Consider the model of an undamped nonlinear spring/mass system given by \(x^{\prime \prime}+8 x-6 x^{3}+x^{5}=0 .\) Use a numerical solver to discuss the nature of the oscillations of the system corresponding to the initial conditions: \(\begin{array}{ll} x(0)=1, x^{\prime}(0)=1 ; & x(0)=-2, x^{\prime}(0)=\frac{1}{2} ; \\ x(0)=\sqrt{2}, x^{\prime}(0)=1 ; & x(0)=2, x^{\prime}(0)=\frac{1}{2} ; \\ x(0)=2, x^{\prime}(0)=0 ; & x(0)=-\sqrt{2}, x^{\prime}(0)=-1 . \end{array}\)

Solve the given system of differential equations by systematic elimination. $$ \begin{aligned} &\frac{d^{2} x}{d t^{2}}+\frac{d y}{d t}=-5 x \\ &\frac{d x}{d t}+\frac{d y}{d t}=-x+4 y \end{aligned} $$

Solve the given differential equation by undetermined coefficients. \(4 y^{\prime \prime}-4 y^{\prime}-3 y=\cos 2 x\)

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

Solve the given differential equation. $$ x^{2} y^{\prime \prime}+3 x y^{\prime}-4 y=0 $$

Solve each differential equation by variation of parameters. $$ y^{\prime \prime}-y=\sinh 2 x $$

Answer Problems \(1-8\) without referring back to the text. Fill in the blank or answer true/false. If simple harmonic motion is described by \(x(t)=\) \((\sqrt{2} / 2) \sin (2 t+\phi)\), the phase angle \(\phi\) is _________ when \(x(0)=-\frac{1}{2}\) and \(x^{\prime}(0)=1\).

In Problems \(1-20\), solve the given system of differential equations by systematic elimination. $$ \begin{aligned} &\frac{d^{2} x}{d t^{2}}+\frac{d y}{d t}=-5 x \\ &\frac{d x}{d t}+\frac{d y}{d t}=-x+4 y \end{aligned} $$