Question

The differential equation of the family of curves $y=e^x(A\cos x+B\sin x)$ where $A,B$ are arbitary constants is

• A. $\frac{d^{2}y}{d{x}^2}-9x=13$
• B. $\frac{d^{2}y}{d{x}^2}-2\frac{dy}{dx}+2y=0$
• C. $\frac{d^{2}y}{d{x}^2}+3y=4$
• D. ${\left(\frac{dy}{dx}\right)}^2+\frac{dy}{dx}-xy=0$

Answer 1

The correct option is B $\frac{d^{2}y}{d{x}^2}-2\frac{dy}{dx}+2y=0$

$y=e^x(A\cos x+B\sin x)y{e}^{-x}=A\cos x+B\sin x-y{e}^{-x}+y_1{e}^{-x}=-A\sin x+B\cos x-y_1{e}^{-x}+y{e}^{-x}+y_2{e}^{-x}-y_1{e}^{-x}=-A\cos x-B\sin x{y}_2{e}^{-x}-2y_1{e}^{-x}+y{e}^{-x}=-y{e}^{-x}\Rightarrow y_2-2y_1+2y=0$