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

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

Answer 2

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