vegetatzz8s

2022-11-25

How can I solve this differential equation? :

SquingonottokQqk

Expert

we have in differential form:

If we put in standard form, and collect terms:

Which is a First Order Separable Ordinary Differential Equation, so we can separate the variables to get:

We can manipulate the RHS integral as follows:

And now both integrals are standard results, so integrating give us:
$\mathrm{ln}|y|=\frac{1}{2}\mathrm{ln}|{x}^{2}+1|+C$
Noting that we require areal solution, and writing $C=\mathrm{ln}A$, we get:
$\mathrm{ln}y=\mathrm{ln}A\sqrt{{x}^{2}+1}$
Giving us the General Solution:
$y=A\sqrt{{x}^{2}+1}$

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