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Problem 2 Easy Difficulty

In Exercises $1-14,$ find the general solution of the differential equation.
$$
\frac{d y}{d x}=\frac{3 x^{2}}{y^{2}}
$$

Answer

The general solution of a differential equation is solving y as a function of x or..

\begin{equation}

y = f(x)

\end{equation}

Separate both variables using multiplication.

\begin{equation}

\frac{dy}{dx} * y^2dx = \frac{3x^2}{y^2} * y^2dx \\

y^2*dy = 3x^2*dx

\end{equation}

Now integrate both sides in order to find the solution.

\begin{equation}

\int y^2\, dy = \int 3x^2\, dx \\

\frac{y^3}{3} = x^3 + C

\end{equation}

Finally, Simplify and solve for y. Remember the constant \(\mathcal{C}\) is needed for indefinite integrals!

\begin{equation}

\frac{y^3}{3}*3 = 3*(x^3+\mathcal{C}) \\
y^3 =3x^3+ \mathcal{C} \\
y = \sqrt[3]{3x^3 + \mathcal{C}}

\end{equation}

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Video Transcript

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