The given differential equation is $x {dy \over dx} = y( \ln y - \ln x +1)$
The initial condition is y(1)=3. Need to find y(3).
How to solve for y(3) if y(1) is known for the given differential equation?
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ordinary-differential-equations
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0You can solve this explicitly. Set $u = \ln(y)$. – 2017-02-02
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0Let $u=\dfrac{y}{x}$. – 2017-02-02
1 Answers
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Let $u=\dfrac{y}{x}$ so given differential equation is \begin{eqnarray} x {dy \over dx} &=& y( \ln y - \ln x +1)\\ u'x+u&=&u(\ln u+1)\\ \dfrac{du}{u\ln u} &=&\dfrac{dx}{x}\\ \int\dfrac{du}{u\ln u} &=&\int\dfrac{dx}{x}\\ \ln\ln u&=&\ln x+\ln C\\ y&=&x3^{x}\\ y(3)&=&81 \end{eqnarray}