Question

Solve the given differential equation by substitution method:

$\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{x}}\mathbf{=}\mathbf{1}\mathbf{+}{\mathit{e}}^{\mathbf{y}\mathbf{-}\mathbf{x}\mathbf{+}\mathbf{5}}$

Short answer

We will solve the given differential equation by substituting.

Step-by-step solution

Step 1:definition of solving differential equation by substitution

Often the first step in solving the differential equation consists of transforming it into another differential equation by means of a substantial.

For example : Suppose we wish to transform the first order differential equation dy/dx=f(x,y) by the substitution y=g(x,u), where u is regardedas a function of the variable x.

Step 2:

$$\begin{gathered} u=y-x+5 \\ \frac{du}{dx}=\frac{dy}{dx}-1 \\ \frac{dy}{dx}=\frac{du}{dx}+1 \end{gathered}$$

Do the substitution,

$$\begin{gathered} \left[\frac{du}{dx}+1\right]=1+e^u \\ \frac{du}{dx}=e^u \end{gathered}$$

Step 3:

Solve by separation of variables

$$\begin{gathered} e^{-u}du=dx \\ \int e^{-u}du=\int dx \\ -e^{-u}=x+C \end{gathered}$$

Step 4:

Substitute back u=y-x+5

$-e^{-\left(y-x+5\right)}=x+C$

Hence, the final result is:

$-e^{x-y-5}=x+C$