$$\left(\frac{\partial f}{\partial y}\right)^2 + + g(y) \frac{\partial f}{\partial y} + h(y) = 0$$
How to solve this differential equation?
Thank you.
Is there a way to solve equation of this form?
0
$\begingroup$
integration
ordinary-differential-equations
derivatives
1 Answers
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We have
$$f'(y)^2 + g(y) f(y) + h(y) = 0$$
Solving the quadratic, $$f'(y) = \frac{-1}{2}g(y) \pm \frac{1}{2}\sqrt{g(y)^2 - 4 h(y)}$$
Integrating both sides,
$$f(y) = C -\frac{1}{2} \int_{y_0}^y g(s) \, ds \pm \frac{1}{2} \int_{y_0}^y \sqrt{g(s)^2 - 4 h(s)} \, ds$$