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I have an ODE: $$xy'= y+2x^3 \sin\big(\frac{y}{x}\big)^2.$$ I have to show that this is separable. I have introduced a function u = y/x. Then using DivideSides and Simplify I got the following equation: $$(ux)'= u+2x^2 \sin\big(\frac{y}{x}\big)^2.$$

Now I have to use product rule for the left side of the equation and get a form of Separable ODE. How can I do that?

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1 Answer 1

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Why not

  1. Substitute $u=y/x$ programmatically:

    x y'[x] == y[x] + 2 x^3*Sin[y[x]/x]^2 /. y -> Function[x, u[x] x] // FullSimplify
    

$$2 x^2 \sin ^2(u(x))=x u'(x)$$

  1. Express the derivative in Leibniz's notation (you can select the output and press Ctrl+Shift+T to transform to pretty TraditionalForm):

    % /. {u[x] -> u, u'[x] -> Dt[u]/Dt[x]}
    

$$2 x^2 \sin ^2(u)=\frac{x du}{dx}$$

  1. Multiply both sides:

     MultiplySides[%, Dt[x]/x/Sin[u]^2, Assumptions -> Dt[x]/x/Sin[u]^2 != 0]
    

$$2 x dx=\csc ^2(u)du$$

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