A solution of the differential equation ( dy dx )^2 - x dy dx + y… - Vidyadip

MCQ

A solution of the differential equation ${\left( {\frac{{dy}}{{dx}}} \right)^2} - x\frac{{dy}}{{dx}} + y = 0$ is

A
$y = 2$
B
$y = 2x$
✓
$y = 2x - 4$
D
$y = 2{x^2} - 4$

✓

Answer

Correct option: C.

$y = 2x - 4$

c
(c) Given equation can be written as $y = x\frac{{dy}}{{dx}} - {\left( {\frac{{dy}}{{dx}}} \right)^2}$

If $\frac{{dy}}{{dx}} = p,$then $y = px - {p^2}$

Differentiating w.r.t. $x$, we get

$p = p + x\frac{{dp}}{{dx}} - 2p\frac{{dp}}{{dx}}$ ==>$\frac{{dp}}{{dx}}(x - 2p) = 0$ ==>$\frac{{dp}}{{dx}} = 0$

Integrating w.r.t. $x$, we get $p = c$

$\therefore \frac{{dy}}{{dx}} = c$; $\therefore y = cx - {c^2}$

If $c = 2$, then $y = 2x - 4$.

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