Question

: Solve a Homogeneous DEq. ydx = 2(x + y) dy

Answer 1

Given: $ydx=2(x+y)dy$

Now $ydx=2(x+y)dy\Rightarrow \frac{dx}{dy}=\frac{2(x+y)}{y}$

Observe that the given differential equation is a first-order homogeneous differential equation.

Solve the equation by using the substitution $x=vy$

$x=vy\Rightarrow \frac{dx}{dy}=v(1)+y(\frac{dv}{dy})=v+y\frac{dv}{dy}$

Then the given differential equation becomes

$\frac{dx}{dy}=\frac{2(x+y)}{y}\Rightarrow v+y\frac{dv}{dy}=\frac{2(vy+y)}{y}=\frac{2y(v+1)}{y}=2(v+1)$

$\Rightarrow y\frac{dv}{dy}=2v+2-v=v+2$

Using the separation of variables, we get

$\frac{dv}{v+2}=\frac{dy}{y}$

Integrating on both sides, we get

$\int \frac{dv}{v+2}=\int \frac{dy}{y}\Rightarrow ln(v+2)=ln(y)+ln(c)\Rightarrow ln(v+2)=ln(cy)$

$\Rightarrow v+2=cy$

Substituting $v=\frac{x}{y}$ , we get

$\frac{x}{y}+2=cy\Rightarrow \frac{x+2y}{y}=cy\Rightarrow x+2y=cy^2$

Therefore, general solution to the given differential equation is

$x+2y=cy^2$