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Looking for a solution to Bernoulli equation

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sky_tm

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Bernoulli equation

\[\frac{{dy}}{{dx}} - \frac{y}{x} = \frac{y^4 cos x}{x^3}\]

Find the general solution.
 

v_c

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Re: Bernoulli equation

The Bernoulli equation takes the form
\[\frac{dy}{dx} + p(x) y = q(x) y^n\]

In your case \[p(x) = -\frac{1}{x}\], \[q(x) = \frac{\cos x}{x^3}\], \[n=4\].

You would start by creating a new variable \[v=y^{1-n}=y^{-3}\],
and follow the formulation given here Bernoulli Differential Equation -- from Wolfram MathWorld

Following the procedure, I am getting
\[y = \frac{x}{\sqrt[3]{-3 \sin x + C}}\]
where \[C\] is a constant. I did not check the result by plugging it back into
the original differential equation yet. Why don't you go through the procedure to see what you get.

I hope this points you in the right direction.

Best regards,
v_c
 

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