Question

Form the differential equation of $y=a{e}^{3x}\cos (x+b)$ Where $y^{\prime }=\frac{dy}{dx}$ and $y^n=\frac{d^{2}y}{d{x}^2}$?

• (A) $y^{\prime \prime }-6y^{\prime }+10y=0$
• (B) $y^{\prime \prime }-6y^{\prime }-10y=0$
• (C) $y^{\prime \prime }+6y^{\prime }-10y=0$
• (D) $y^{\prime \prime }+6y^{\prime }+10y=0$

Answer 1

The Correct Option is C

Explanation:
Given,$y=a{e}^{3x}\cos (x+b)$There are two constants a and b so differentiate two times Differentiating wr.t $x$,We get,$y^{\prime }=3a{e}^{3x}\cos (x+b)-a{e}^{3x}\sin (x+b)$$\Rightarrow y^{\prime }=3y-a{e}^{3x}\sin (x+b)$$\Rightarrow a{e}^{3x}\sin (x+b)=3y-y^{\prime }$Differentiating again w.r.t $x$,We get,$3{ae}^{3x}\sin (x+b)+a{e}^{3x}\cos (x+b)=3y^{\prime }-y^{\prime \prime }$$\Rightarrow 3(3y-y^{\prime })+y-3y^{\prime }-y^n=0$$\Rightarrow 9y-3y^{\prime }+y-3y^{\prime }-y^n=0$$\Rightarrow y^{\prime \prime }+6y^{\prime }-10y=0$Hence, the correct option is (C).