Today a classmate of mine had given me and my teacher a differential equation I don't know from where but it became a headache for me and my teacher because we both were unable to solve it at some particular step. Here is the differential equation:
$$x\, \frac{dy}{dx} + y \ln(x) = x\,e^x\,x^{(1/2) \ln x}.$$
When I and my teacher was solving we compared it by $dy/dx + Py =Q, $ but when it came to integrate $Q$ (I.F.) it would integrable by me. So please help me in this question. Any guidance will be appreciated.
There might be a $+/-$ mistake: if either '$+ y \,\text{ln}(x)$' is changed to '$- y \,\text{ln}(x)$' or $x^{\frac{1}{2}\,\text{ln}(x)}$ is changed to $x^{-\frac{1}{2}\,\text{ln}(x)}$, the resulting integral is explicitly solvable. As it stands, the integral, which is essentially $\int e^{x + \text{ln}(x)^2}\,\text{d}x$, is not solvable.