DramaQueen001
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Consider the differential equation [tex] x^{2} \frac{dy}{dx} =6 y^{2} +6xy[/tex] which may be considered either as a homogenous equation or as a Bernoulli equation.
If we make the substitution [tex]y(x)=xv(x)[/tex] relevant to homogenous equations, we obtain [tex] \frac{dv}{dx}=[/tex]
If we make the substitution [tex]z(x)= (y(x))^{-1} [/tex] relevant to homogenous equations, we obtain [tex] \frac{dz}{dx}=[/tex]
Using either (or both) of these methods, solve the initial value problem for the above equation where y(3)=6. Find the interval of validity of this solution.

Respuesta :

LammettHash
As a Bernoulli equation:

[tex]x^2\dfrac{\mathrm dy}{\mathrm dx}=6y^2+6xy\iff x^2y^{-2}\dfrac{\mathrm dy}{\mathrm dx}-6xy^{-1}=6[/tex]

Let [tex]z=y^{-1}\implies\dfrac{\mathrm dz}{\mathrm dx}=-y^{-2}\dfrac{\mathrm dy}{\mathrm dx}[/tex]. The ODE becomes

[tex]-x^2\dfrac{\mathrm dz}{\mathrm dx}-6xz=6[/tex]
[tex]x^6\dfrac{\mathrm dz}{\mathrm dx}+6x^5z=-6x^4[/tex]
[tex]\dfrac{\mathrm d}{\mathrm dx}[x^6z]=-6x^4[/tex]
[tex]x^6z=-6\displaystyle\int x^4\,\mathrm dx[/tex]
[tex]x^6z=-\dfrac65x^5+C[/tex]
[tex]z=-\dfrac6{5x}+\dfrac C{x^6}[/tex]
[tex]y^{-1}=-\dfrac6{5x}+\dfrac C{x^6}[/tex]
[tex]y=\dfrac1{\frac C{x^6}-\frac6{5x}}[/tex]
[tex]y=\dfrac{5x^6}{C-6x^5}[/tex]

With [tex]y(3)=6[/tex], we get

[tex]6=\dfrac{5(3)^6}{C-6(3)^5}\implies C=\dfrac{4131}2[/tex]

so the solution is

[tex]y=\dfrac{5x^6}{\frac{4131}2-6x^5}=\dfrac{10x^6}{4131-12x^5}[/tex]

which is valid as long as the denominator is not zero, which is the case for all [tex]x\neq\sqrt[5]{\dfrac{4131}{12}}[/tex].

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