The general solution of the differential equation, y ' + y ' (x) … - Vidyadip

MCQ

The general solution of the differential equation, $y ' + y\phi ' (x) - \phi (x) . \phi ’(x) = 0$ where $\phi (x)$ is a known function is : where $c$ is an arbitrary constant .

✓
$y = ce^{-\phi (x)} + \phi (x) - 1$
B
$y = ce^{+\phi (x)} + \phi (x) - 1$
C
$y = ce^{-\phi (x)} -\phi (x) + 1$
D
$y = ce^{-\phi (x)} + \phi (x) + 1$

✓

Answer

Correct option: A.

$y = ce^{-\phi (x)} + \phi (x) - 1$

a
$\frac{{dy}}{{dx}} \,+ \, y \phi '(x) = \phi (x).\phi '(x)$
$I.F. = {e^{\int {\phi '(x)dx} }} = {e^{\phi (x)}}$
hence $y.e^{\phi (x)} = e^{\phi (x)}.\phi (x).\phi '(x) dx = e^t.t dt$
where $\phi (x) = t$
$= te^t - e^t + C = \phi (x).e^{\phi (x)} - e^{\phi (x)} + C$
$\therefore y = ce^{-\phi (x)} + \phi (x) - 1$

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