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9. differential equations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths9. differential equations1 Mark
MCQ
The solution of the equation $\sqrt {a + x} \frac{{dy}}{{dx}} + x = 0$ is
  • ✓
    $3y + 2\sqrt {a + x} .(x - 2a) = 3c$
  • B
    $3y + 2\sqrt {x + a} .(x + 2a) = 3c$
  • C
    $3y + \sqrt {x + a} .(x + 2a) = 3c$
  • D
    None of these
✓

Answer

Correct option: A.
$3y + 2\sqrt {a + x} .(x - 2a) = 3c$
a
(a) $\sqrt {a + x} \frac{{dy}}{{dx}} + x = 0$ ==> $\int_{}^{} {dy} = - \int_{}^{} {\frac{x}{{\sqrt {a + x} }}dx} $

==> $y = - \int_{}^{} {\sqrt {a + x} } dx + \int_{}^{} {\frac{a}{{\sqrt {a + x} }}} dx$
$\left\{ \because \int_{{}}^{{}}{\frac{x}{\sqrt{a+x}}}dx=\int_{{}}^{{}}{\frac{x+a-a}{\sqrt{a+x}}}dx \right\}$

==> $y = - \frac{2}{3}{(a + x)^{3/2}} + 2a\sqrt {a + x} + c$

==> $3y = - \sqrt {a + x} (2(a + x) - 6a) + 3c$

==> $3y = - 2\sqrt {a + x} (x - 2a) + 3c$

==> $3y + 2\sqrt {a + x} (x - 2a) = 3c$.

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