Question

A solution of the differential equation \( 2x^2\dfrac{d^2y}{dx^2} + 3x\dfrac{dy}{dx} - y = 0, \, x > 0 \) that passes through the point (1, 1) is

• A. \( y = \dfrac{1}{x} \)
• B. \( y = \dfrac{1}{x^2} \)
• C. \( y = \dfrac{1}{\sqrt{x}} \)
• D. \( y = \dfrac{1}{x^{3/2}} \)

Hint:

For Cauchy–Euler equations, always try \( y = x^m \) substitution to convert it into an algebraic equation in \( m \).

Answer 1

The Correct Option is A

To solve this problem, we need to find a solution to the differential equation:

\(2x^2\dfrac{d^2y}{dx^2} + 3x\dfrac{dy}{dx} - y = 0, \, x > 0\)

Given that it passes through the point (1, 1), we need to test the appropriate solution among the given options.

Let's substitute each of the given options into the differential equation to check which satisfies it:

1. Option 1: \(y = \dfrac{1}{x}\)
   • First derivative: \(\dfrac{dy}{dx} = -\dfrac{1}{x^2}\)
   • Second derivative: \(\dfrac{d^2y}{dx^2} = \dfrac{2}{x^3}\)
   • Substitute into the differential equation:

\(2x^2\left(\dfrac{2}{x^3}\right) + 3x\left(-\dfrac{1}{x^2}\right) - \dfrac{1}{x} = 0\)

Simplifying: \(=\dfrac{4}{x} - \dfrac{3}{x} - \dfrac{1}{x} = 0\)

• This satisfies the differential equation. Therefore, Option 1 is correct.

1. Option 2: \(y = \dfrac{1}{x^2}\)
   • First derivative: \(\dfrac{dy}{dx} = -\dfrac{2}{x^3}\)
   • Second derivative: \(\dfrac{d^2y}{dx^2} = \dfrac{6}{x^4}\)
   • Substitute into the differential equation:

\(2x^2\left(\dfrac{6}{x^4}\right) + 3x\left(-\dfrac{2}{x^3}\right) - \dfrac{1}{x^2} \neq 0\)

• Option 2 does not satisfy the differential equation.

1. Option 3: \(y = \dfrac{1}{\sqrt{x}}\)
   • First derivative: \(\dfrac{dy}{dx} = -\dfrac{1}{2x^{3/2}}\)
   • Second derivative: \(\dfrac{d^2y}{dx^2} = \dfrac{3}{4x^{5/2}}\)
   • Substitute into the differential equation:

\(2x^2\left(\dfrac{3}{4x^{5/2}}\right) + 3x\left(-\dfrac{1}{2x^{3/2}}\right) - \dfrac{1}{\sqrt{x}} \neq 0\)

• Option 3 does not satisfy the differential equation.

1. Option 4: \(y = \dfrac{1}{x^{3/2}}\)
   • First derivative: \(\dfrac{dy}{dx} = -\dfrac{3}{2x^{5/2}}\)
   • Second derivative: \(\dfrac{d^2y}{dx^2} = \dfrac{15}{4x^{7/2}}\)
   • Substitute into the differential equation:

\(2x^2\left(\dfrac{15}{4x^{7/2}}\right) + 3x\left(-\dfrac{3}{2x^{5/2}}\right) - \dfrac{1}{x^{3/2}} \neq 0\)

• Option 4 does not satisfy the differential equation.

After testing all options, only \(y = \dfrac{1}{x}\) satisfies the differential equation, and it also passes through the point (1, 1). Thus, the correct answer is:

\(y = \dfrac{1}{x}\)