Let the solution y = y(x) of the differential equation dy/dx - y = 1 + 4sinx satisfy y(π) = 1.

Question

Let the solution \(y=y(x)\) of the differential equation \(\frac{d y}{d x}-y=1+4 \sin x\) satisfy \(y(\pi)=1\). Then \(y\left(\frac{\pi}{2}\right)+10\) is equal to _______.

Answer 1

Correct answer: 7

\( e^{-x}=\int\left(e^{-x}+4 e^{-x} \sin x\right) d x\)

\(\mathrm{ye}^{-\mathrm{x}}=-\mathrm{e}^{-\mathrm{x}}-2\left(\mathrm{e}^{-\mathrm{x}} \sin \mathrm{x} \mathrm{e}^{-\mathrm{x}} \cos \mathrm{x}\right)+\mathrm{C}\)

\(\mathrm{y}=-1-2(\sin \mathrm{x}+\cos \mathrm{x})+\mathrm{ce}^{\mathrm{x}}\)

\(\because \mathrm{y}(\pi)=1 \Rightarrow \mathrm{c}=0\)

\(y(\pi / 2)=-1-2=-3\)

Now,

\(y(\pi / 2) + 10\)

\(= (-3) + 10\)

\(= 7\)