Differential Equations
Order and Degree
MMTS_Full_Test_17
Grade 12
Question:
The differential equation formed by eliminating constants $A$ and $B$ from $y=A\cos(\ln x)+B\sin(\ln x)$ is
$x^2\dfrac{d^2y}{dx^2}+x\dfrac{dy}{dx}+y=0$
$x^2\dfrac{d^2y}{dx^2}-x\dfrac{dy}{dx}+y=0$
$x^2\dfrac{d^2y}{dx^2}+x\dfrac{dy}{dx}-y=0$
$\dfrac{d^2y}{dx^2}+y=0$
Step-by-Step Solution
Key Concept: Cauchy-Euler ODE
$y'=\frac{-A\sin(\ln x)+B\cos(\ln x)}{x}$. $xy'=-A\sin(\ln x)+B\cos(\ln x)$. $(xy')'=y'+xy''=\frac{-A\cos(\ln x)-B\sin(\ln x)}{x}=-y/x$. So $xy''+y'+y/x=0\Rightarrow x^2y''+xy'+y=0$.
Correct Answer: 1