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Q.
A solution of the differential equation $\bigg (\frac {dy}{dx} \bigg )^2-x \frac {dy}{dx}+ y=0 $ is
AIEEEAIEEE 1999Differential Equations
Solution:
Given differential equation is
$\bigg ( \frac {dy}{dx} \bigg )^2 -x \frac {dy}{dx} + y=0 ...(i)$
(a) $y=2 \Rightarrow \frac {dy}{dx} =0 $
On putting in Eq. (i), $0^2-x(0)+y=0$
$\Rightarrow y=0$ which is not satisfied.
(b) $ y=2x \, \Rightarrow \frac {dy}{dx}=2$
On putting in Eq. (i),
$(2)^2-x.2+y=0$
$\Rightarrow 4-2x+y=0$
$\Rightarrow y=2x $ which is not satisfied
(c) $y=2x-4 \Rightarrow \frac {dy}{dx}=2$
On putting in Eq. (i)
$(2)^2-2x+2x-4=0$
$4 - 2x + 2x -4 = 0\,\,[\because y = 2x - 4]$
$y=2x-4$ is satisfied.
(d) $y=2x^2-4$
$\frac {dy}{dx}=4x$
On putting in Eq. (i),
$(4x)^2-x.4x+y=0$
$\Rightarrow y=0 $ which is not satisfied.