Q.1 Determine whether y = (c - e ^ x)/(2x); y^ prime =- 2y+e^ x 2x is a solution for the differential equation Q.2 Solve the Initial value problem ln(y ^ x) * (dy)/(dx) = 3x ^ 2 * y given y(2) = e ^ 3 . Q.3 Find the general solution for the given differential equation. (dy)/(dx) = (2x - y)/(x - 2y)

(Q.1)

Then substituting into the DE gives

and both sides match, so y is indeed a valid solution.

(Q.2)

This DE is separable, since you can write . So you have

Integrate both sides (on the left, the numerator suggests a substitution):

Given y (2) = e ³, we find

so that the particular solution is

(Q.3) I believe I've already covered in another question you posted.

answer: x=60