First order nonlinear differential equation help?

Integrate both sides and get y⁵ = xy + c.

So x = (y⁵ - c)/y.

x '' + 3x ' - 10x = t e^(t) The homogeneous eqn is r^2 + 3r - 10 = 0 => (r + 5)(r - 2) = 0 r = 2 and -5 the established answer is x(0) = A e^(2t) + Be^(-5t) enable particular answer is xp = (Ct + D)e^t x ' = Ce^t + (Ct + D)e^t = e^t (Ct + C + D) x '' = e^t (Ct + C + D) + Ce^t = e^t (Ct + 2C + D) substituting interior the given DE e^t (Ct + 2C + D) + 3e^t(Ct + C + D) - 10e^t(Ct + D) = te^t e^t [Ct + 2C + D + 3Ct + 3C + 3-D - 10Ct - 10D ] = t e^t e^t [ -6Ct + 5C - 6D ] = t e^t te^t(-6C) + 5C - 6D = te^t -6C = a million ==> C = -a million/6 5C - 6D = 0 -6D = 5/6 D = -5/36 xp =[ (-a million/6)t - 5/36) ]e^t = - (a million/6)e^t - (5/36)e^t x(t) = x(0) + xp x(t) = A e^(2t) + Be^(-5t) - (a million/6)e^t - (5/36)e^t