Question

Verify that y = e² ͯ sin x is a solution to the differential equation
d²y/dx² - 4 dy/dx + 5y = 0

Answer 1

$\text{since we have }\\ y=e^{2x} \ sin x\\ \text{we get}\\ y^\prime=2e^{2x} \ sin \ x+e^{2x} \ cos \ x\\ y ^{\prime\prime}=4e^{2x} \ sin \ x+2 e^{2x} \ cos \ x\\ +2 e^{2x} \ cos \ x-e^{2x} \ sin \ x\rightarrow(1)\\ \text{Also, we have}\\ -4y^\prime=-8e^{2x} \ sin \ x-4e^{2x} \ cos \ x\rightarrow (2)\\ 5 y=5e^{2x} \ sin \ x\rightarrow (3)\\ \text{Adding (1), (2), (3), we get }\\ y^{\prime \prime}-4y^\prime+5y=0$