Question

dy/dx = 3 y x^2 and y=3 when x=-2, then what is y?

Answer 1

3

Answer 2

It's a separable equation.

Answer 3

can you draw it?

Answer 4

\[
\begin{array}{rcl}
\frac{dy}{dx} &=& 3 y x^2 \\
\frac{1}{y} \frac{dy}{dx} &=& 3 x^2 \\
\int \frac{1}{y} \frac{dy}{dx}dx &=& \int 3 x^2dx \\
u=y&&du = \frac{dy}{dx}dx \\
\int \frac{1}{u} du &=& \int 3 x^2dx \\
\ln(u) +C_2 &=& x^3+C_1 \\
\ln(y) +C_2 &=& x^3+C_1 \\
\ln(y) &=& x^3+C_3 \\
e^{\ln(y)} &=& e^{x^3+C_3} \\
y &=& Ce^{x^3} \\
\end{array}
\]Then you have to plug in to solve for \(C\).

Answer 5

What happen to the 1/3?

Answer 6

Oh nvm haha

Answer 7

\[
\int x^2 dx = \frac{x^3}{3}+C
\]

Answer 8

We solve for \(C\):\[
\begin{array}{rcl}
y &=& Ce^{x^3} \\
(3) &=& Ce^{(-2)^3} \\
3 &=& Ce^{-8} \\
3e^{8} &=& C \\
\end{array}
\]

Giving us, in the end: \[
y = (3e^8)e^{x^3} = 3e^{x^3+8}
\]

Answer 9

@jenny0828 Get it?

Answer 10

Very thorough, wio. I appreciate your help!