Question

Solve differential equation: \[ x^2 \frac{d^2y}{dx^2} + 4x \frac{dy}{dx} + 2y = 0, x \geq 1 \] with initial conditions $y=0, \; y'(1)=1$ at $x=1$. Find $y$ at $x=2$. (round off to two decimals)

Hint:

For Euler–Cauchy equations, always try $y=x^m$. Roots of auxiliary equation give solution powers. Apply conditions to find constants.

Answer 1

Step 1: Recognize type.
Equation is Cauchy–Euler form: \[ x^2 y'' + 4x y' + 2y = 0 \]

Step 2: Trial solution.
Assume $y = x^m$. Then: \[ m(m-1)x^m + 4mx^m + 2x^m = 0 \] \[ m^2 + 3m + 2 = 0 \] \[ (m+1)(m+2)=0 \Rightarrow m=-1,-2 \]

Step 3: General solution.
\[ y(x) = \frac{A}{x} + \frac{B}{x^2} \]

Step 4: Apply conditions.
At $x=1$, $y(1)=0$: \[ A + B = 0 \Rightarrow B = -A \] Derivative: \[ y'(x) = -\frac{A}{x^2} - \frac{2B}{x^3} \] At $x=1$, $y'(1)=1$: \[ - A - 2B = 1 \] Substitute $B=-A$: \[ - A - 2(-A) = -A + 2A = A = 1 \] So $A=1, B=-1$.

Step 5: Final solution.
\[ y(x) = \frac{1}{x} - \frac{1}{x^2} \] At $x=2$: \[ y(2) = \frac{1}{2} - \frac{1}{4} = 0.25 \] \[ \boxed{0.25} \]