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DIFFERENTIAL EQUATIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDIFFERENTIAL EQUATIONS1 Mark
Question
State True or False for the following:
The solution of the differential equation $\frac{\text{dy}}{\text{dx}}=\frac{\text{x}+2\text{y}}{\text{x}}$ is $\text{x}+\text{y}=\text{k}\text{x}^2.$
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Answer

True.Solution:
We have
$\frac{\text{dy}}{\text{dx}}=\frac{\text{x}+2\text{y}}{\text{x}}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=1+\frac{2}{\text{x}}.\text{y}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}-\frac{2}{\text{x}}=\text{y}$
This is a linear differential equation.
$\therefore\text{I.F.}=\text{e}^{\frac{-2}{\text{x}}\text{dx}}$
$=\text{e}^{-2\log\text{x}}=\text{x}^{-2}$
Thus, the differential solution is given as,
$\text{y}.\text{x}^{-2}=\int\text{x}^{-2}.1\text{dx}+\text{k}$
$\Rightarrow\frac{\text{y}}{\text{x}^2}=\frac{\text{x}^{-2+1}}{-2+1}+\text{k}$
$\Rightarrow\frac{\text{y}}{\text{x}^2}=\frac{-1}{\text{x}}+\text{k}$
$\Rightarrow\text{y}=-\text{x}+\text{k}\text{x}^2$
$\Rightarrow\text{y}+\text{x}=\text{k}\text{x}^2$

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