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DIFFERENTIAL EQUATIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDIFFERENTIAL EQUATIONS2 Marks
Question
For each of the differential equations given in find the general solution:
$\frac{\text{dy}}{\text{dx}}+\frac{\text{y}}{\text{x}}=\text{x}^2$
✓

Answer

The given differential equation is:$\frac{\text{dy}}{\text{dx}}+​​\text{py}=\text{Q }{(\text{where p}}=\frac{1}{\text{x}}\text{and Q}=\text{x}^2\big)$
$\text{Now, I.F}=\text{e}^{\int\text{pdx}}=\text{e}^{\int\frac{1}{\text{x}}\text{dx}}=\text{e}^{\log\text{x}}=\text{x}.$
The solution of the given differential equation is given by the relation,
$\text{y}(\text{I.F})=\int(\text{Q}\times\text{I.F.})\text{dx}+\text{C}$
$\Rightarrow\ \text{y}\ (\text{x})=\int(\text{x}^2\cdot\text{x})\text{dx}+\text{C}$
$\Rightarrow\ ​​​​\text{xy}=\int\text{x}^3\text{dx}+\text{C}$
$\Rightarrow\ \text{xy}=\frac{\text{x}^4}{4}+\text{C}$
This is the required general solution of the given differential equation.

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