91Ó°ÊÓ

Verify that the indicated function is an explicit solution of the given differential equation. Assume an appropriate interval \(I\) of definition for each solution. $$ \frac{d y}{d t}+20 y=24 ; \quad y=\frac{6}{5}-\frac{6}{5} e^{-20 t} $$

In Problems 7-12, match each of the given differential equations with one or more of these solutions: (a) \(y=0\), (b) \(y=2\) (c) \(y=2 x\) (d) \(y=2 x^{2}\). $$ x y^{\prime \prime}-y^{\prime}=0 $$

y=c_{1} e^{x}+c_{2} e^{-x}\( is a two-parameter family of solutions of the second-order DE \)y^{\prime \prime}-y=0$. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions. $$ y(1)=0, \quad y^{\prime}(1)=e $$

Verify that the indicated function is an explicit solution of the given differential equation. Assume an appropriate interval \(I\) of definition for each solution. $$ y^{\prime \prime}-6 y^{\prime}+13 y=0 ; \quad y=e^{3 x} \cos 2 x $$

In Problems 13 and 14, determine by inspection at least one solution of the given differential equation. $$ y^{\prime \prime}=y^{\prime} $$

y=c_{1} e^{x}+c_{2} e^{-x}\( is a two-parameter family of solutions of the second-order DE \)y^{\prime \prime}-y=0$. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions. $$ y(-1)=5, \quad y^{\prime}(-1)=-5 $$

Verify that the indicated function is an explicit solution of the given differential equation. Assume an appropriate interval \(I\) of definition for each solution. $$ y^{\prime \prime}+y=\tan x ; \quad y=-(\cos x) \ln (\sec x+\tan x) $$

y=c_{1} e^{x}+c_{2} e^{-x}\( is a two-parameter family of solutions of the second-order DE \)y^{\prime \prime}-y=0$. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions. $$ y(0)=0, \quad y^{\prime}(0)=0 $$

In Problems 13 and 14, determine by inspection at least one solution of the given differential equation. $$ y^{\prime}=y(y-3) $$

$$ y^{\prime}=3 y^{2 / 3}, \quad y(0)=0 $$