Question

Let \( y(x) = x v(x) \) be a solution of the differential equation \[ x^2 \frac{d^2y}{dx^2} - 3x \frac{dy}{dx} + 3y = 0. \] If \( v(0) = 0 \) and \( v(1) = 1, \) then \( v(-2) \) is equal to .................

Hint:

When solving differential equations, always look for substitutions or transformations that simplify the equation, such as \( y(x) = x v(x) \) in this case.

Answer 1

Correct Answer: 4

Step 1: Substitute \( y(x) = x v(x) \) \text{ into the given differential equation.}
We begin by substituting \( y(x) = x v(x) \) into the equation: \[ x^2 \frac{d^2}{dx^2} (x v(x)) - 3x \frac{d}{dx} (x v(x)) + 3x v(x) = 0. \] We calculate the derivatives: \[ \frac{d}{dx}(x v(x)) = v(x) + x \frac{dv}{dx}, \quad \frac{d^2}{dx^2}(x v(x)) = 2 \frac{dv}{dx} + x \frac{d^2v}{dx^2}. \] Substituting these into the differential equation and simplifying, we get: \[ x^2 (2 \frac{dv}{dx} + x \frac{d^2v}{dx^2}) - 3x (v(x) + x \frac{dv}{dx}) + 3x v(x) = 0. \]
Step 2: Simplify and solve for \( v(x) \).
After simplifying and solving the equation, we find that the solution for \( v(x) \) satisfies: \[ v(x) = \frac{1}{x}. \]
Step 3: Compute \( v(-2) \).
Since \( v(x) = \frac{1}{x} \), we have: \[ v(-2) = \frac{1}{-2} = -\frac{1}{2}. \]
Step 4: Conclusion.
Therefore, \( v(-2) \) is: \[ \boxed{-\frac{1}{2}}. \]