The solution of the differential equation d2ydx2=exsin⁡x when y1π4=0,yπ2=0 is

dydx=∫exsin⁡xdx=exsin⁡(x−π/4)2+C1⇒y=12exsin⁡(x−π/2)2+C1x+C20=yπ2=0+C1π2+C20=y1π4=12eπ/4−sin⁡π4+cos⁡π/4+C1⇒C1=0 So C2=0Hence y=12exsin⁡x−π2.

The solution of the differential equation d2ydx2=exsin⁡x when y1π4=0,yπ2=0 is

dydx=∫exsin⁡xdx=exsin⁡(x−π/4)2+C1⇒y=12exsin⁡(x−π/2)2+C1x+C20=yπ2=0+C1π2+C20=y1π4=12eπ/4−sin⁡π4+cos⁡π/4+C1⇒C1=0 So C2=0Hence y=12exsin⁡x−π2.

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The solution of the differential equation
$\frac{d^{2} y}{d x^{2}} = e^{x} \sin x$ when $y_{1} (\frac{π}{4}) = 0, y (\frac{π}{2}) = 0$ is

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$y = e^{x} \sin (x - π / 4) + x + \frac{π}{2}$

$y = (1 / 2) e^{x} \sin (x + π / 2)$

$y = (1 / \sqrt{2}) e^{x} \sin (x - π / 2)$

$y = (1 / 2) e^{x} \sin (x - π / 2)$

answer is D.

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Detailed Solution

$\begin{array}{l} \frac{d y}{d x} = \int e^{x} \sin x d x = \frac{e^{x} \sin (x - π / 4)}{\sqrt{2}} + C_{1} \\ \Rightarrow y = \frac{1}{\sqrt{2}} \frac{e^{x} \sin (x - π / 2)}{\sqrt{2}} + C_{1} x + C_{2} \\ 0 = y (\frac{π}{2}) = 0 + C_{1} \frac{π}{2} + C_{2} \\ 0 = y_{1} (\frac{π}{4}) = \frac{1}{2} e^{π / 4} [- \sin \frac{π}{4} + \cos π / 4] + C_{1} \\ \Rightarrow C_{1} = 0 So C_{2} = 0 \end{array}$