Solve the differential equation Differential Equations Miscellaneous Exercise 12.

Name: Solve the differential equation Differential Equations Miscellaneous Exercise 12.
Uploaded: Mon, 2019-06-10 18:16
Description: Solve the differential equation Differential Equations Miscellaneous Exercise 12.

Q12. Solve the differential equation $\left[\frac{e^{-2\sqrt x}}{\sqrt x} - \frac{y}{\sqrt x} \right ]\frac{dx}{dy} = 1\; \ (x\neq 0)$ .

Answers (1)

Given,
$[\frac{e^{- 2 \sqrt{x}}}{\sqrt{x}} - \frac{y}{\sqrt{x}}] \frac{d x}{d y} = 1$
$\Rightarrow \frac{dy}{dx} = \frac{e^{- 2 \sqrt{x}}}{\sqrt{x}} - \frac{y}{\sqrt{x}}$
$⟹ \frac{dy}{dx} + \frac{y}{\sqrt{x}} = \frac{e^{- 2 \sqrt{x}}}{\sqrt{x}}$
This is equation is in the form of $\frac{d y}{d x} + p y = Q$
$p = \frac{1}{\sqrt{x}} and Q = \frac{e^{- 2 \sqrt{x}}}{\sqrt{x}}$
Now, I.F. $= e^{\int pdx} = e^{\int \frac{1}{\sqrt{x}} dx} = e^{2 \sqrt{x}}$
We know that the solution of the given differential equation is:
$y (I \cdot F .) = \int (Q \cdot F \cdot) d x + C$
$\Rightarrow {ye}^{2 \sqrt{x}} = \int (\frac{e^{- 2 \sqrt{x}}}{\sqrt{x}} \times e^{2 \sqrt{x}}) dx + C$
$\Rightarrow {ye}^{2 \sqrt{x}} = \int \frac{1}{\sqrt{x}} dx + C$
$\Rightarrow {ye}^{2 \sqrt{x}} = 2 \sqrt{x} + C$

Posted by

HARSH KANKARIA

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Q12. Solve the differential equation .

Answers (1)

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HARSH KANKARIA

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Q12. Solve the differential equation $\left[\frac{e^{-2\sqrt x}}{\sqrt x} - \frac{y}{\sqrt x} \right ]\frac{dx}{dy} = 1\; \ (x\neq 0)$ .