Question

Solve the given differential equation by using appropriate Substitution;

$\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{x}}\mathbf{=}\frac{\mathbf{1}\mathbf{-}\mathbf{x}\mathbf{-}\mathbf{y}}{\mathbf{x}\mathbf{+}\mathbf{y}}$

Short answer

We will solve the given differential equation by substituting.

Step-by-step solution

Step 1:

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 regarded as a function of the variable x.

solving the following by substitution;

$$\begin{gathered} \frac{dy}{dx}=\frac{1-\left(x+y\right)}{x+y} \\ \frac{du}{dx}-1=\frac{1-u}{u} \\ letu=x+y \\ \frac{du}{dx}=\frac{dy}{dx}-1 \\ \frac{du}{dx}=\frac{1-u}{u}+1 \\ \frac{du}{dx}=\frac{1-u}{u}+\frac{u}{u} \\ \frac{du}{dx}=\frac{1}{u} \end{gathered}$$

finding the integral solution

$$\begin{gathered} udu=dx \\ \int udu=\int dx \\ \frac{1}{2}u=x+C \\ \frac{1}{2}{\left(x+y\right)}^2=x+C \\ Hencethefinalansweris\frac{1}{2}{\left(x+y\right)}^2=x+C \end{gathered}$$