Question

• Express \(y\) in terms of \(x\) for differential equation \(\dfrac{dy}{dx}=y\ln y\)

Solution

Same steps for differential equations.

$$ \begin{align*} \frac{1}{y\ln y}dy&=dx\\ \int\frac{1}{y\ln y}dy&=\int dx\\ \end{align*} $$

Let \(u=\ln y\), and notice \(\displaystyle\int dx=x\).

$$ \begin{align*} \int\frac{1}{u}dy&=\int dx\\ \ln|u|&=x+C\\ \ln|\ln y|&=x+C\\ y&=ke^{e^x} \end{align*} $$