First slide

Methods of solving first order first degree differential equations

Question

Let y=y(x) be the solution curve of the differential equation $(y^{2} - x) \frac{d y}{d x} = 1$ , satisfying y(0) = 1. This curve intersects the x-axis at a point whose abscissa is:

Moderate

A

$2 - e$
B

$2 + e$
C

2
D

$- e$

Solution

We have $\frac{d x}{d y} + x = y^{2}$
I.F. = $e^{\int 1 d y} = e^{y}$
$∴$ Solution is
$x . e^{y} = \int y^{2} . e^{y} . d y$
$= y^{2} . e^{y} - \int 2 y . e^{y} . d y$
$= y^{2} e^{y} - 2 (y . e^{y} - e^{y}) + c$
$∴ x . e^{y} = y^{2} e^{y} - 2 y e^{y} + 2 e^{y} + C$
$\Rightarrow x = y^{2} - 2 y + 2 + c . e^{- y}$
Given $x = 0, y = 1$
$\Rightarrow 0 = 1 - 2 + 2 + \frac{c}{e}$
$\Rightarrow c = - e$
Now. curve cuts x-axis $\Rightarrow y = 0$
$\Rightarrow x = 0 - 0 + 2 + (- e) (e^{- 0})$
$\Rightarrow x = 2 - e$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App