Section C.2 Visualizing Solutions

Verification Details for Visualization DE.

To verify that

y = C e^{x^{2}} + 3

is a solution, we move all terms to one side of the equation to get

\frac{d y}{d x} - 2 x y + 6 x = 0.

Now, substitute it into the differential equation:

\begin{aligned} \frac{d y}{d x} - 2 x y + 6 x = & 0 \\ \frac{d}{d x} (C e^{x^{2}} + 3) - 2 x (C e^{x^{2}} + 3) + 6 x = & 0 \\ 2 C x e^{x^{2}} - 2 C x e^{x^{2}} - 6 x + 6 x = & 0 \\ 0 = & 0 \end{aligned}

This shows that

y = C e^{x^{2}} + 3

satisfies and is, thus, a solution to

\frac{d y}{d x} = 2 x y - 6 x .

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