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DIFFERENTIAL EQUATIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDIFFERENTIAL EQUATIONS5 Marks
Question
For each of the differential equation given in find the general solution:
$\text{x}\frac{\text{dy}}{\text{dx}}+2\text{y}=\text{x}^2\log\text{x}$
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Answer

The given differential equation is:
$\text{x}\frac{\text{dy}}{\text{dx}}+2\text{y}=\text{x}^2\log\text{x}$
$\Rightarrow\ \ \frac{\text{dy}}{\text{dx}}+\frac{2}{\text{x}}\text{y}=\text{x}\log\text{x}$
The equation is in the form of a linear differential equation as:
$\frac{\text{dy}}{\text{dx}}+\text{py}=\text{Q}\ \big(\text{where p}=\frac{2}{\text{x}}\ \text{and Q}=\text{x}\log\text{x}\big)$
$\text{Now, I.F}=\text{e}^{\int\text{pdx}}=\text{e}^{\int\frac{2}{\text{x}}\text{dx}}=\text{e}^{2\log\text{x}}=\text{e}^{\log\text{x}^2}=\text{x}^2.$
The general solution of the given differential equation is given by the relation,
$\text{y}(\text{I.F})=\int(\text{Q}\times\text{I.F})\text{dx}+\text{C}$
$\Rightarrow\ \ \text{y}\ \cdot​​\text{x}^2=\int(\text{x}\log\text{x}\cdot\text{x}^2)\text{dx}+\text{C}$
$\Rightarrow\ \text{x}^2\text{y}=\int(\text{x}^3\log\text{x})\text{dx}+\text{C}$
$\Rightarrow\ \text{x}^2\text{y}=\log\text{x}\cdot\int\text{x}^3 \text{dx}-\int\bigg[\frac{\text{d}}{\text{dx}}(\log\text{x})\cdot\int\text{x}^3\ \text{dx}\bigg]\text{dx}+\text{C}$
$\Rightarrow\ \text{x}^2\text{y}=\log\text{x}\cdot\frac{\text{x}^4}{4}-\int\bigg(\frac{1}{\text{x}}\cdot\frac{\text{x}^4}{4}\bigg)\text{dx}+\text{C}$
$\Rightarrow\ \text{x}^2\text{y}=\frac{\text{x}^{4}\log\text{x}}{4}-\frac{1}{4}\int\text{x}^3\text{dx}+\text{C}$
$\Rightarrow\ \text{x}^2\text{y}=\frac{\text{x}^{4}\log\text{x}}{4}-\frac{1}{4}\cdot\frac{\text{x}^4}{4}+\text{C}$
$\Rightarrow\ \text{x}^2\text{y}=\frac{1}{16}\text{x}^2(4\log\text{x}-1)+\text{Cx}$
$\Rightarrow\ \text{y}=\frac{1}{16}\text{x}^2(4\log\text{x}-1)+\text{Cx}^{-2}$

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