Question

Consider the following homogeneous differential equation. y dx = 2(x + y) dy Use the substitution x = vy to write the given differential equation in terms of only y and v.

Answer 1

Answer:

$ydv = (v +2)dy$            

Step-by-step explanation:

We are given the following differential equation:

$y dx = 2(x + y) dy$

We have to substitute

$x = vy$

Differentiating we get,

$\dfrac{dx}{dy} = v + y\dfrac{dv}{dy}$

Putting value in differential equation, we get,

$y dx = 2(x + y) dy\\\\y\dfrac{dx}{dy}=2(x+y)\\\\y(v+y\dfrac{dv}{dy}) = 2(vy + y)\\\\vy + y^2\dfrac{dv}{dy} = 2vy +2y\\\\y^2\dfrac{dv}{dy}=vy +2y\\\\y^2dv = y(v+2)dy\\ydv = (v +2)dy$

is the differential equation after substitution.