Question

Let y'=4 x. find all values of r such that y = rx^{2} satisfies the differential equation. if there is more than one correct answer, enter your answers as a comma separated list.

Answer 1

First we solve the differential equation:
 y '= 4 x
 dy / dx = 4 x
 dy = 4x * dx
 Integrating both sides we have
 int (dy) = int (4x * dx)
 y = 4 (x^2/2)
 y = 2x^2
 Therefore, comparing both functions:
 y = 2x ^ 2
 y = rx ^ 2
 We conclude that
 r = 2
 answer
 The value of r that satisfies the differential equation is
 r = 2

Answer 2

Y' = dy/dx = 4x 
To obtain y we integrate wrt x, so y = 4 int (x)  
y = 4 x^2/2 = 2x^2 
But y = rx^2 
So 2x^2 = rx^2 
Comparing coefficients we find that r = 2