4.1.1.0.1 Example 1

\begin{align} x^{\prime }\left ( t\right ) +y^{\prime }\left ( t\right ) & =x+y+t\tag {1}\\ x^{\prime }\left ( t\right ) +y^{\prime }\left ( t\right ) & =2x+3y+e^{t} \tag {2}\end{align}

Hence

\begin{align} x+y+t & =2x+3y+e^{t}\nonumber \\ y & =-\frac {1}{2}x+\frac {1}{2}t-\frac {1}{2}e^{t} \tag {3}\end{align}

Taking derivative w.r.t. \(t~\) gives

\begin{equation} y^{\prime }=-\frac {x^{\prime }}{2}+\frac {1}{2}-\frac {1}{2}e^{t} \tag {4}\end{equation}

Substituting (3,4) in (1) to eliminate \(y,y^{\prime }\) gives

\begin{align} x^{\prime }+\left ( -\frac {x^{\prime }}{2}-\frac {1}{2}e^{t}+\frac {1}{2}\right ) & =x+\left ( -\frac {x}{2}-\frac {1}{2}e^{t}+\frac {1}{2}t\right ) +t\nonumber \\ x^{\prime } & =3t+x-1 \tag {5}\end{align}

This is linear ode. Its solution is

\begin{equation} x=c_{1}e^{t}-3t-2 \tag {6}\end{equation}

Substituting this in (3) gives

\begin{align*} y & =-\frac {1}{2}\left ( c_{1}e^{t}-3t-2\right ) +\frac {1}{2}t-\frac {1}{2}e^{t}\\ & =2t-\frac {1}{2}e^{t}-\frac {1}{2}c_{1}e^{t}+1 \end{align*}