Question

D.E whose solution is $y=e^{3x}(c_1\cos x+c_2\sin x)$ :

• A. $y_2-3y_1+5y=0$
• B. $y_2-6y_1+10y=0$
• C. $y_2+6y_1+10y=0$
• D. $y_2-6y_1-10y=0$

Answer 1

The correct option is B $y_2-6y_1+10y=0$

$y=e^{3x}(c_1\cos x+c_2\sin x)$
$\Rightarrow y_1=e^{3x}(-c_1\sin x+c_2\cos x)+3e^{3x}(c_1\cos x+c_2\sin x)$
$\Rightarrow y_1-3y=e^{3x}(-c_1\sin x+c_2\cos x)$
$\Rightarrow y_2-3y_1=3e^{3x}(-c_1\sin x+c_2\cos x)+e^{3x}(-c_1\cos x-c_2\sin x)$
$\Rightarrow (y_2-3y_1)=3(y_1-3y)+(-y)$
$\Rightarrow y_2-6y_1+10y=0$