Name: Formation of Differential Equations
Uploaded: 2019-07-19
Duration: 11 min 9 s
Description: In this video, we are going to learn about the formation of differential equations in detail. Master this concept in just a few minutes with this quick lesson.

Question

Reduce the differential equation

x y \frac{d^{2} y}{d x^{2}} - x {(\frac{d y}{d x})}^{2} + y \frac{d y}{d x} = 0

into a first order equation using the substitution

\frac{d y}{d x} = u y

. Hence or otherwise, find its general solution.

Accepted Answer

$x y \frac{d^{2} y}{d x^{2}} - x {(\frac{d y}{d x})}^{2} + y \frac{d y}{d x} = 0 \dots (i)$

Let $\frac{d y}{d x} = u y \dots (ii)$

Differentiating w.r.t. $x$ ,

$\frac{d^{2} y}{d x^{2}} = \frac{u d y}{d x} + y \frac{d u}{d x} \dots (iii)$

from $(i)$ , $(ii)$ and $(iii)$ , we have

$x y (\frac{u d y}{d x} + y \frac{d u}{d x}) - x u^{2} y^{2} + y \cdot u y = 0$

$u x y \frac{d y}{d x} + x y^{2} \frac{d u}{d x} - x u^{2} y^{2} + u y^{2} = 0$

$\Rightarrow u x y \cdot u y + x y^{2} \frac{d u}{d x} - x u^{2} y^{2} + u y^{2} = 0$

$x y^{2} \frac{d u}{d x} + u y^{2} = 0$

$x \frac{d u}{d x} = - u \Rightarrow x \frac{d u}{d x} + u = 0 . . . (iv)$

$\Rightarrow \frac{d u}{u} = \frac{- d x}{x}$

Integrating both sides w.r.t. $x$

$\Rightarrow \int \frac{d u}{u} = - \int \frac{d x}{x}$

$\Rightarrow \log u = - \log x + \log C$

$\Rightarrow u = \frac{C}{x} \Rightarrow \frac{1}{y} \frac{d y}{d x} = \frac{C}{x}$

$\Rightarrow \int \frac{d y}{y} = C \cdot \int \frac{d x}{x}$

$\Rightarrow \log y = C \log x + \log k$

$\Rightarrow y = k x^{C}$

Reduce the differential equation $x y \frac{d^{2} y}{d x^{2}} - x {(\frac{d y}{d x})}^{2} + y \frac{d y}{d x} = 0$ into a first order equation using the substitution $\frac{d y}{d x} = u y$ . Hence or otherwise, find its general solution.

Important Questions on Differential Equations - II

Learn and Prepare for any exam you want !

Reduce the differential equation xyd2ydx2-xdydx2+ydydx=0 into a first order equation using the substitution dydx=uy. Hence or otherwise, find its general solution.

Important Questions on Differential Equations - II

Learn and Prepare for any exam you want !

Reduce the differential equation $x y \frac{d^{2} y}{d x^{2}} - x {(\frac{d y}{d x})}^{2} + y \frac{d y}{d x} = 0$ into a first order equation using the substitution $\frac{d y}{d x} = u y$ . Hence or otherwise, find its general solution.