d-2-y-dx-2-sin-3x-e-x-x-2-when-y-0-1-and-y-0-0-Find-the-solution-

Question Number 133776 by bramlexs22 last updated on 24/Feb/21

\frac{d^{2} y}{{dx}^{2}} = \sin 3 x + e^{x} + x^{2}, when y^{'} (0) = 1

and y (0) = 0 . Find the solution

Answered by bobhans last updated on 24/Feb/21

\frac{d}{d x} [\frac{d y}{d x}] = \sin 3 x + e^{x} + x^{2}

d [\frac{d y}{d x}] = (\sin 3 x + e^{x} + x^{2}) d x

\int d [\frac{d y}{d x}] = \int (\sin 3 x + e^{x} + x^{2}) d x

\Rightarrow \frac{d y}{d x} = - \frac{1}{3} \cos 3 x + e^{x} + \frac{1}{3} x^{3} + C_{1}

y^{'} (0) = - \frac{1}{3} + 1 + C_{1} = 1; C_{1} = \frac{1}{3}

\int d y = \int (- \frac{1}{3} \cos 3 x + e^{x} + \frac{1}{3} x^{3} + \frac{1}{3}) d x

y = - \frac{1}{9} \sin 3 x + e^{x} + \frac{1}{12} x^{4} + \frac{1}{3} x + C_{2}

y (0) = 1 + C_{2} = 0 \Rightarrow C_{2} = - 1

∴ y = - \frac{\sin 3 x}{9} + e^{x} + \frac{x^{4} + 4 x - 12}{12}

Facebook Tweet Pin