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Ivenika [448]
3 years ago
7

X dy/dx +2y =x^2logx by using Bernoulli's Equation

Mathematics
1 answer:
inna [77]3 years ago
3 0
This ODE isn't of Bernoulli type, but it is linear, so we should be able to find an integrating factor to solve it.

x\dfrac{\mathrm dy}{\mathrm dx}+2y=x^2\log x\implies\dfrac{\mathrm dy}{\mathrm dx}+\dfrac2xy=x\log x

The integrating factor will be

\mu(x)=\exp\left(\displaystyle\int\frac2x\,\mathrm dx\right)=x^2

Multiplying both sides of the ODE by the IF gives

x^2\dfrac{\mathrm dy}{\mathrm dx}+2xy=x^3\log x
\dfrac{\mathrm d}{\mathrm dx}[x^2y]=x^3\log x
x^2y=\displaystyle\int x^3\log x\,\mathrm dx

Integrate the right hand side by parts to get

x^2y=\dfrac14x^4\log x-\dfrac1{16}x^4+C
y=\dfrac14x^2\log x-\dfrac1{16}x^2+\dfrac C{x^2}
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