91Ó°ÊÓ

In Problems 13-16, 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. In calculus, the curvature of a curve that is defined by a function \(y=f(x)\) is defined as $$ \kappa=\frac{y^{\prime \prime}}{\left[1+\left(y^{\prime}\right)^{2}\right]^{3 / 2}} $$ Find \(y=f(x)\) for which \(\kappa=1\). [Hint: For simplicity, ignore constants of integration.]

Solve the given differential equation. $$ x y^{(4)}+6 y^{\prime \prime \prime}=0 $$

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

In Problems 11-20, find the eigenvalues and eigenfunctions for the given boundary-value problem. $$ y^{\prime \prime}+2 y^{\prime}+(\lambda+1) y=0, y(0)=0, y(5)=0 $$

In Problems 1-26, solve the given differential equation by undetermined coefficients. $$ y^{\prime \prime}-2 y^{\prime}+5 y=e^{x} \cos 2 x $$

Determine whether the given set of functions is linearly dependent or linearly independent on the interval \((-\infty, \infty)\). $$ f_{1}(x)=5, \quad f_{2}(x)=\cos ^{2} x, \quad f_{3}(x)=\sin ^{2} x $$

In Problems, find the eigenvalues and eigenfunctions for the given boundary- value problem. $$ y^{\prime \prime}+2 y^{\prime}+(\lambda+1) y=0, y(0)=0, y(5)=0 $$

Solve the given differential equation by undetermined coefficients. \(y^{\prime \prime}-2 y^{\prime}+5 y=e^{x} \cos 2 x\)

Solve the given system of differential equations by systematic elimination. $$ \begin{aligned} &D x=y \\ &D y=z \\ &D z=x \end{aligned} $$

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