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

Step 1: Recognize the Cauchy-Euler equation.
The equation is second-order Cauchy-Euler. Try \(y = x^m\).

Step 2: Substitute.
\[ m(m-1) + 3m - 3 = 0 \implies m^2 + 2m - 3 = 0 \implies (m+3)(m-1) = 0 \]
So \(m = 1\) or \(m = -3\).

Step 3: General solution.
\[ y(x) = C_1x + C_2x^{-3} \]

Step 4: Apply initial conditions.
\(y(1)=3\): \(C_1 + C_2 = 3\). \(y'(x) = C_1 - 3C_2x^{-4}\), \(y'(1)=-5\): \(C_1 - 3C_2 = -5\).
Subtracting: \(4C_2 = 8 \Rightarrow C_2 = 2\), so \(C_1 = 1\).

Step 5: Evaluate at x=2.
\[ y(x) = x + 2x^{-3}, \quad y(2) = 2 + 2/8 = 2.25 \]

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