Chapter 4: Higher-Order Differential Equations

In Problemsthe indicated function${\mathit{y}}_{\mathbf{1}}\mathbf{\left(}\mathit{x}\mathbf{\right)}$is a solution of the given differential equation. Use reduction of order or formula(5),as instructed, to find a second solution${\mathit{y}}_{\mathbf{2}}\mathbf{\left(}\mathit{x}\mathbf{\right)}$

$\left(1-x^2\right)\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathit{x}\mathit{y}\mathbf{\text{'}}\mathbf{=}\mathbf{0}\mathbf{;}\mathbf{}{\mathit{y}}_{\mathbf{1}}\mathbf{=}\mathbf{1}$.

Solve the given differential equation.${\mathit{x}}^{\mathbf{3}}\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathit{x}\mathit{y}\mathbf{\text{'}}\mathbf{-}\mathit{y}\mathbf{=}\mathbf{0}$

In Problems 15–26 find a linear differential operator that annihilates

the given function.

${\mathbf{x}}^{\mathbf{3}}\mathbf{(}\mathbf{1}\mathbf{-}\mathbf{5}\mathbf{x}\mathbf{)}$

In Problems, 15-28 find the general solution of the given higher-order differential equation.

$\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathit{y}\mathbf{=}\mathbf{0}$

In Problems 13–18 proceed as in Example 3 to find a solution of the

given initial-value problem. Evaluate the integral that defines \({y_p}\left( x \right)\).

16.

Solve the given system of differential equations by systematic elimination.

$\begin{array}{rcl}{D}^{2}x-2\left(D^2+D\right)y& =& sint\\ x+Dy& =& 0\end{array}$

Show that the substitution u = y’ leads to a Bernoulli equation. Solve this equation

xy''=y'+x(y')²

In Problems 15 and 16 find a homogeneous second-order Cauchy Euler equation with real coefficients if the given numbers are roots of its auxiliary equation.
${\mathit{m}}_{\mathbf{1}}\mathbf{=}\mathit{i}$

In Problemsthe indicated function${\mathit{y}}_{\mathbf{1}}\mathbf{\left(}\mathit{x}\mathbf{\right)}$is a solution of the associated homogeneous equation. Use the method of reduction of order to find a second solution${\mathit{y}}_{\mathbf{2}}\mathbf{\left(}\mathit{x}\mathbf{\right)}$of the homogeneous equation and a particular solution ${\mathit{y}}_{\mathit{P}}\mathbf{\left(}\mathit{x}\mathbf{\right)}$of the given nonhomogeneous equation.

$\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{4}\mathit{y}\mathbf{=}\mathbf{2}\mathbf{;}\mathbf{}{\mathit{y}}_{\mathbf{1}}\mathbf{=}{\mathit{e}}^{\mathbf{-}\mathbf{2}\mathbf{x}}$

In Problems 15–26 find a linear differential operator that annihilates

the given function.

$\mathbf{1}\mathbf{+}\mathbf{7}{\mathbf{e}}^{\mathbf{2}\mathbf{x}}$