Mathematics: Write the equation in the form y′=f(y/x) then use the substitution y=xu to find an implicit general solution. Then solve the initial value problem. y′=5y2+4x2xy, y(1)=3 The resulting differential equation in x x and u u can be written as xu′= which is separable. Separating variables we arrive at du=dx/x Separating variables and and simplifying the solution can be written in the form u2+1=Cf(x) where C is an arbitrary constant and f(x)= . Transforming back into the variables x and y and using the

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Write the equation in the form y′=f(y/x) then use

crispingolfer7082

21.01.2022

Looks like the ODE is supposed to be

$y'=\dfrac{5y^2+4x^2}{xy}$

In that case, let y(x)=xu(x) , so that y'=xu'+u . Substitute these into the ODE to get

$xu'+u=\dfrac{5x^2u^2+4x^2}{x^2u}$

$xu'=\dfrac{4u^2+4}u$

which is separable as

$\dfrac u{u^2+1}\,\mathrm du=4x\,\mathrm dx$

Integrate both sides to get

$\dfrac12\ln(u^2+1)=2x^2+C$

We can solve for u^2 :

$\ln(u^2+1)=4x^2+C$

$u^2+1=Ce^{4x^2}$

$u^2=Ce^{4x^2}-1$

Replace $u=\frac yx$ to get the general solution,

$\dfrac{y^2}{x^2}=Ce^{4x^2}-1$

Given that y(1)=3 , we find

$9=Ce^4-1\implies C=10e^{-4}$

so the particular solution is

$\dfrac{y^2}{x^2}=10e^{4(x^2-1)}-1$

$\implies y=\pm x\sqrt{10e^{4(x^2-1)}-1}$

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