Gujarat BoardEnglish MediumSTD 12 ScienceMathsDifferential Equations3 Marks
Question
Find the general solution of $\frac{d y}{d x}+\frac{y}{x}=x^{2}$
✓
Answer
It is given that $\frac{d y}{d x}+\frac{y}{x}=x^{2}$ This is equation in the form of $\frac{d y}{d x}+p y=Q$ (where, p = $\frac{1}{x}$ and Q = x2) Now, I.F. = $\mathrm{e}^{\int \mathrm{pdx}}=\mathrm{e}^{\int \frac{1}{\mathrm{x}} \mathrm{dx}}=\mathrm{e}^{\log \mathrm{x}}=\mathrm{x}$ Thus, the solution of the given differential equation is given by the relation: $y(\mathrm{I} . \mathrm{F} .)=\int(\mathrm{Q} \times \mathrm{I} . \mathrm{F} .) \mathrm{d} \mathrm{x}+\mathrm{C}$ $\Rightarrow \mathrm{y}(\mathrm{x})=\int\left(\mathrm{x}^{2} \cdot \mathrm{x}\right) \mathrm{d} \mathrm{x}+\mathrm{C}$ $\Rightarrow x y=\int\left(x^{3}\right) d x+C$ $\Rightarrow x y=\frac{x^{4}}{4}+C$ Therefore, the required general solution of the given differential equation is $x y=\frac{x^{4}}{4}+C$
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