Question

# Solve the following initial value problems an.d leave the solution in implicit form. Use graphing software to plot the solution.. If the implicit solution. describes more than one function., be sure to indicate which fun.ct ion. corresponds to the solution. of the initial value problem. $y^{\prime}(t)=\frac{2 t^{2}}{y^{2}-1}, y(0)=0$

Solution

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It is given that

$y^\prime(t)=\frac{2t^2}{y^2-1}\,\, , \,\, y(0)=0$

To solve the initial value problem we will first transform the equation above into its separable form:

\begin{align*} \frac{dy}{dt}&=\frac{2t^2}{y^2-1}\\ (y^2-1)\, dy&=2t^2\, dt\tag{\footnotesize\textcolor{#c34632}{integrate both sides}}\\ \int (y^2-1)\, dy&=2\int t^2\, dt\\ \int y^2\, dy-\int \, dy&=2\cdot \frac{t^3}{3}+C\\ \frac{y^3}{3}-y&=\frac{2}{3}\cdot t^3+C \\ y^3-3y&=2t^3+C \end{align*}

To find the constant $C$ we will use the given initial condition $y(0)=0$:

$y(0)=0\Rightarrow 0=0+C\Rightarrow C=0$

We conclude that the solution to the initial value problem is

$\pmb{y^3-3y=2t^3}$

The plot of the curve described by the solution in implicit form is given bellow.

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