Question

In Problems 35–38, $\mathit{y}\mathbf{=}{\mathit{c}}_{\mathbf{1}}{\mathit{e}}^{\mathbf{3}\mathbf{x}}\mathbf{+}{\mathit{c}}_{\mathbf{2}}{\mathit{e}}^{\mathbf{-}\mathbf{x}}\mathbf{-}\mathbf{2}\mathit{x}$is a two-parameter family of the second-order DE $\mathit{y}\mathbf{-}\mathbf{2}\mathit{y}\mathbf{\text{'}}\mathbf{-}\mathbf{3}\mathit{y}\mathbf{=}\mathbf{6}\mathit{x}\mathbf{+}\mathbf{4}$. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions.

$\mathit{y}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{=}\mathbf{4}\mathbf{,}\mathbf{}\mathit{y}\mathbf{\text{'}}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{=}\mathbf{-}\mathbf{2}$

Short answer

$y=\frac{3}{2}{e}^{3x-3}+\frac{9}{2}{e}^{1-x}-2x$

Step-by-step solution

Taking derivative with respect to x and value of one parameter

Given: $y=c_1{e}^{3x}+c_2{e}^{-x}-2x······\left(1\right)$

Take the derivative with respect to x,

$y\text{'}=3c_1{e}^{3x}-c_2{e}^{-x}-2······\left(2\right)$

Initial condition in (1),

$\begin{array}{c}{c}_1{e}^3+c_2{e}^{-1}-2=4\\ c_1{e}^3=6-c_2{e}^{-1}\\ c_2{e}^{-1}=6-c_1{e}^3\\ c_1{e}^3=6-c_2{e}^{-1}\end{array}$

Initial condition in (2) and Substitute $c_1{e}^3=6-c_2{e}^{-1}$, we get

$\begin{array}{c}3c_1{e}^3-c_2{e}^{-1}-2=-2\\ 3c_1{e}^3-(6-c_1{e}^3)-2=-2\\ 3c_1{e}^3-6+c_1{e}^3=0\\ 4c_1{e}^3=6\end{array}$

Simplify, we get

$c_1=\frac{3}{2}{e}^{-3}$

Find the value of another parameter and solution of the second order IVP

Substitute $c_1=\frac{3}{2}{e}^{-3}$in the equation (2),

$\begin{array}{c}{c}_2{e}^{-1}=6-\frac{3}{2}{e}^{-3}{e}^3\\ c_2{e}^{-1}=\frac{9}{2}\\ c_2=\frac{9}{2}e\end{array}$

Substitute the values of $c_1$and $c_2$in the given equation (1) we get,

$\begin{array}{c}y=\frac{3}{2}{e}^{-3}{e}^{3x}+\frac{9}{2}e{e}^{-x}-2x\\ y=\frac{3}{2}{e}^{3x-3}+\frac{9}{2}{e}^{1-x}-2x\end{array}$

Therefore, the solution of the second order IVP consisting of this differential equation and the given initial conditions is $y=\frac{3}{2}{e}^{3x-3}+\frac{9}{2}{e}^{1-x}-2x$.