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Section C.2 Visualizing Solutions
Verification Details for Visualization DE.
To verify that \(\ds y = Ce^{x^2} + 3\) is a solution, we move all terms to one side of the equation to get
\begin{equation*}
\ds \frac{dy}{dx} - 2xy + 6x = 0.
\end{equation*}
Now, substitute it into the differential equation:
\begin{align*}
\frac{dy}{dx} - 2xy + 6x =\amp\ 0\\
\frac{d}{dx}\left( Ce^{x^2} + 3 \right) - 2x\left( Ce^{x^2} + 3 \right) + 6x =\amp\ 0\\
2Cxe^{x^2} - 2Cxe^{x^2} - 6x + 6x =\amp\ 0\\
0 =\amp\ 0
\end{align*}
This shows that \(\ds y = Ce^{x^2} + 3\) satisfies and is, thus, a solution to \(\ds \frac{dy}{dx} = 2xy - 6x.\)
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