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DIFFERENTIAL EQUATIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDIFFERENTIAL EQUATIONS3 Marks
Question
Solve the differential equation $\bigg[\frac{\text{e}^{-2\sqrt{\text{x}}}}{\sqrt{\text{x}}}-\frac{\text{y}}{\sqrt{\text{x}}}\bigg]\frac{\text{dx}}{\text{dy}}=1(\text{x}\neq0).$
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Answer

Given: Differential equation: $\bigg(\frac{\text{e}^{-2\sqrt{\text{x}}}}{\sqrt{\text{x}}}-\frac{\text{y}}{\sqrt{\text{x}}}\bigg)\frac{\text{dx}}{\text{dy}}=1$
$\Rightarrow\ \Big(\frac{\text{e}^{-2\sqrt{\text{x}}}}{\sqrt{\text{x}}}-\frac{\text{y}}{\sqrt{\text{x}}}\Big)\frac{\text{dx}}{\text{dy}}=1$ $\Rightarrow\ \ \frac{\text{dy}}{\text{dx}}+\frac{\text{y}}{\sqrt{\text{x}}}=\frac{\text{e}^{-2\sqrt{\text{x}}}}{\sqrt{\text{x}}}$
$\text{Comparing this equation with }\frac{\text{dy}}{\text{dx}}+\text{Py}=\text{Q},$ $\text{P}=\frac{1}{\sqrt{\text{x}}}\ \text{and}\ \text{Q}=\frac{\text{e}^{-2\sqrt{\text{x}}}}{\sqrt{\text{x}}}$
$\int\text{P}\ \text{dx}=\int\frac{1}{\sqrt{\text{x}}}\text{dx}=\int\text{x}^{\frac{-1}{2}}\text{dx}=\frac{\text{x}^{\frac{1}{2}}}{\frac{1}{2}}=2\sqrt{\text{x}}$ $\text{I.F}=\text{e}^{\int\text{pdx}}=\text{e}^{2\sqrt{\text{x}}}$
$\text{The general solution is}\ \ \text{y}(\text{I.F})=\int\text{Q}(\text{I.F})\text{dx}+\text{c}$
$\Rightarrow\ \ \text{y}^{2\sqrt{\text{x}}}=\int\frac{\text{e}^{-2\sqrt{\text{x}}}}{\sqrt{\text{x}}}\text{e}^{2\sqrt{\text{x}}}\ \text{dx}+\text{c}=\int\frac{1}{\sqrt{\text{x}}}\text{dx}+\text{x}$
$\Rightarrow\ \ \text{y}^{2\sqrt{\text{x}}}=\int\text{x}^{\frac{-1}{2}}\ \text{dx}+\text{c}=\frac{\text{x}^{\frac{1}{2}}}{\frac{1}{2}}+\text{c}=2\sqrt{\text{x}}+\text{c}$ $\Rightarrow\ \ \text{y}=\text{e}^{-2\sqrt{\text{x}}}\ (2\sqrt{\text{x}}+\text{c})$

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