Question

Consider a differential equation:
\[ x^2 \frac{d^2y}{dx^2} + 3x\frac{dy}{dx} - 3y = 0 \]
with \(y = 3\) and \(\dfrac{dy}{dx} = -5\) at \(x = 1\).
The value of \(y\) at \(x = 2\) is ______ (rounded off to two decimal places).

Hint:

This is a Cauchy-Euler equation; substitute \(y=x^m\) to get the characteristic equation \(m^2+2m-3=0\).

Answer 1

Correct Answer: 2.25

Using the substitution $x=e^t$: $x^2y'' + 3xy' - 3y=0$ becomes $\dfrac{d^2y}{dt^2}+2\dfrac{dy}{dt}-3y=0$, with auxiliary equation $s^2+2s-3=0$ giving $s=1,-3$. So $y(x)=Ax+Bx^{-3}$. At $x=1$ ($t=0$): $A+B=3$, and $A-3B=-5$. Solving gives $B=2$, $A=1$. So $y(2)=2+2/8=2.25$.

\[ \boxed{y(2) = 2.25} \]