Q1Advanced Engineering Mathematics I

Equation: $(D^2+9)y=\cos 2t$

1. Complementary Function (CF): Auxiliary equation: $m^2+9=0⟹m=\pm 3i$. $CF=C_1\cos 3t+C_2\sin 3t$.

2. Particular Integral (PI): $PI=\frac{1}{D^2+9}\cos 2t$ Replace $D^2$ with $-2^2=-4$: $PI=\frac{1}{-4+9}\cos 2t=\frac{1}{5}\cos 2t$.

3. General Solution: $y(t)=C_1\cos 3t+C_2\sin 3t+\frac{1}{5}\cos 2t$.

4. Apply Boundary Conditions: $y(0)=1⟹C_1+\frac{1}{5}=1⟹C_1=4/5$. $y(\pi /2)=-1⟹C_1\cos (3\pi /2)+C_2\sin (3\pi /2)+\frac{1}{5}\cos (\pi )=-1$ $0-C_2-1/5=-1⟹C_2=4/5$.

Final solution: $y(t)=\frac{4}{5}\cos 3t+\frac{4}{5}\sin 3t+\frac{1}{5}\cos 2t$.