The solution y ( x ) of the differential equation d x 2 d 2 y = sin 3 x + e x + x 2 when y 1 ( 0 ) = 1 and y ( 0 ) = 0 is

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The solution y ( x ) of the differential equation d x 2 d 2 y ​ = sin 3 x + e x + x 2 when y 1 ​ ( 0 ) = 1 and y ( 0 ) = 0 is

TestHub · Accepted Answer

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

Mathematics - Differential Equation Question with Solution | TestHub

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