The solution yx of the differential equation d2 y-dx2 - sin 3x - ex - x2 when y1 0 - 1 and y0 - 0 is

The correct option is A sin 3x/9 + e^x + x^4/12 + 1/3 x 1Integrating the given differential equation, we have dy/dx = cos 3x/3 + e^x + x^3/3 + C_1 but y ...

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Standard XII

Mathematics

Variable Separable Method

The solution ...

Question

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

- \frac{s i n 3 x}{9} + e^{x} + \frac{x^{4}}{12} + \frac{1}{3} x - 1

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- \frac{s i n 3 x}{9} + e^{x} + \frac{x^{4}}{12} + \frac{1}{3} x

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- \frac{c o s 3 x}{3} + e^{x} + \frac{x^{4}}{12} + \frac{1}{3} x + 1

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Solution

The correct option is A $- \frac{s i n 3 x}{9} + e^{x} + \frac{x^{4}}{12} + \frac{1}{3} x - 1$
Integrating the given differential equation, we have $\frac{d y}{d x} = - \frac{c o s 3 x}{3} + e^{x} + \frac{x^{3}}{3} + C_{1}$
but $y_{1} (0)$ 1 so $1 = - \frac{1}{3} + 1 + C_{1} \Rightarrow C_{1} = \frac{1}{3}$
Again integrating, we get $y = - \frac{s i n 3 x}{9} + e^{x} + \frac{x^{4}}{12} + \frac{1}{3} x + C_{2}$
but $y (0) = 0$ so $0 = 1 + C_{2} \Rightarrow C_{2} = - 1$ . Thus $y = - \frac{s i n 3 x}{9} + e^{x} + \frac{x^{4}}{12} + \frac{1}{3} x - 1$

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The solution y(x) of the differential equation d2ydx2=sin 3x+ex+x2 when y1(0)=1 and y(0) = 0 is

The correct option is A −sin 3x9+ex+x412+13x−1Integrating the given differential equation, we have dydx=−cos 3x3+ex+x33+C1 but y1(0) 1 so 1=−13+1+C1⇒C1=13 Again integrating, we get y=−sin 3x9+ex+x412+13x+C2 but y(0)=0 so 0=1+C2⇒C2=−1. Thus y=−sin 3x9+ex+x412+13x−1

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