Differential Equation and Applications (p-1) — Maths (commerce) STD 12 Commerce / Arts — Question
Maharashtra BoardEnglish MediumSTD 12 Commerce / ArtsMaths (commerce)Differential Equation and Applications (p-1)1 Mark
Question
Solve the following differential equations:$e^{d y / d x}=x$
✓
Answer
$ \begin{gathered} e^{d y / d x}=x \quad \therefore \frac{d y}{d x}=\log x \\ \therefore d y=\log x d x \quad \therefore \int 1 d y=\int \log x d x \\ \text { Now } \int \log x d x=\int(\log x)(1) d x \quad \ldots \ldots \ldots \text { (1) } \\ =(\log x) \int 1 d x-\int\left[\frac{d}{d x}(\log x) \cdot \int 1 d x\right] d x \\ \\ =(\log x)(x)-\int \frac{1}{x} \cdot x d x=x \log x-\int 1 d x \\ =x \log x-x \end{gathered} $ $\therefore$ from (1), the general solution is $ y=x \log x-x+c, \text { i.e. } y=x(\log x-1)+c $
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