First slide

Methods of solving first order first degree differential equations

Question

$The solution of the differential equation \frac{d^{2} y}{d x^{2}} = \sin 3 x + e^{x} + x^{2} when y^{1} (0) = 1 and y(0)=0 is$

Moderate

A

$- \frac{\sin 3 x}{9} + e^{x} + \frac{x^{4}}{12} + \frac{1}{3} x - 1$
B

$- \frac{\sin 3 x}{9} + e^{x} + \frac{x^{4}}{12} + \frac{1}{3} x$
C

$- \frac{\cos 3 x}{9} + e^{x} + \frac{x^{4}}{12} + \frac{1}{3} x + 1$
D

$\frac{\cos 3 x}{9} + \frac{x^{4}}{12} + \frac{x}{3} - e^{x} + 1$

Solution

$Given that \frac{d^{2} y}{d x^{2}} = \sin 3 x + e^{x} + x^{2}$
Integrating on both sides
$\frac{d y}{d x} = - \frac{cos 3 x}{3} + e^{x} + \frac{x^{3}}{3} + c$
$And given y^{1} (0) = 1$
$\Rightarrow 1 = - \frac{1}{3} + 1 + 0 + C$
$\Rightarrow C = - \frac{1}{3}$
$∴ \frac{d y}{d x} = - \frac{cos 3 x}{3} + e^{x} + \frac{x^{3}}{3}$
Again Integrating on both sides
$y = - \frac{sin 3 x}{9} + e^{x} + \frac{x^{4}}{12} - \frac{x}{3} + K$
And given y (0) = 0
$\Rightarrow 0 = 0 + 1 + 0 - 0 + K$
$∴ K = - 1$
$∴ y = - \frac{\sin 3 x}{9} + e^{x} + \frac{x^{4}}{12} - \frac{x}{3} - 1$

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