Simplify. x(dy/dx) - 3y - x^6 e^(x^3) , y(0) = 2

Question

Simplify.

x (d y / d x) - 3 y - x^{6} e x^{3}, y (0) = 2

Answer 1

\frac{d y}{d x} - \frac{3 y}{x} = x^{5} e^{x^{3}}

linear diff. eqn.
y' +P(x) y = Q(x)
P(x)=-3/y

Q (x) = x^{5} \exp (x^{3})

i (x) = \exp (\int \frac{- 3}{x} d x) = \exp (- 3 \ln x) = \exp (\ln x^{- 3}) = \frac{1}{x^{3}}

y = \int \frac{1}{x^{3}} x^{5} e^{x^{3}} d x + c

y = \int x^{2} e^{x^{3}} d x + c

x^{3} = u

3 x^{2} d x = d u

x^{2} d x = d u / 3

y = \frac{1}{3} \int e^{u} d u + c

y = \frac{e^{x^{3}}}{3} + c

y = \frac{x^{3} e^{x^{3}}}{3} + c x^{3}

Answers (1)