Question

11. The differential equation of the form dy/dx=ax+by+c/dx+ey+f
(A) equation reduce to homogeneous form
(B) variable separable
(C) exact
(D) none of these​

Answer 1

A.equation reduce to homogeneous form

Answer 2

(A) equation reduces to homogeneous form

The differential equation of the form $\frac{dy}{dx}=\frac{ax+by+c}{dx+ey+f}$ reduces to homogeneous form.

Explanation:

• Let us understand the given condition with an example.

Let us consider the following differential equation,

$\quad\frac{dy}{dx}=\frac{x+y+1}{x+y-1}$

Put $x+y=v$ so that

$\quad\frac{dv}{dx}=1+\frac{dy}{dx}$

Hence we obtain,

$\quad \frac{dv}{dx}-1=\frac{v+1}{v-1}$

$\Rightarrow \frac{dv}{dx}=\frac{v+1+v-1}{v-1}$

$\Rightarrow \frac{dv}{dx}=\frac{2v}{v-1}$

$\Rightarrow \frac{v-1}{v}dv=2dx$

$\Rightarrow dv-\frac{dv}{v}=2dx$

On integration, we get

$\quad \int dv-\int \frac{dv}{v}=2\int dx+c$ where $c$ is constant of integration

$\Rightarrow v-log(v)=2x+c$

$\Rightarrow x+y-log(x+y)=2x+c$ since $v=x+y$

$\Rightarrow y-log(x+y)=x+c$ which is the required solution.