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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 $\displaystyle\int dx=x$.

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