91Ó°ÊÓ

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

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.

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

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

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

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

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

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

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

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: \(\begin{array}{ll} x(0)=1, x^{\prime}(0)=1 ; & x(0)=-2, x^{\prime}(0)=\frac{1}{2} ; \\ x(0)=\sqrt{2}, x^{\prime}(0)=1 ; & x(0)=2, x^{\prime}(0)=\frac{1}{2} ; \\ x(0)=2, x^{\prime}(0)=0 ; & x(0)=-\sqrt{2}, x^{\prime}(0)=-1 . \end{array}\)