Question

A complete solution of \( y'' + a_1 y' + a_2 y = 0 \) is \( y = b_1 e^{-x} + b_2 e^{-3x} \), where \( a_1, a_2, b_1 \) and \( b_2 \) are constants, then the respective values of \( a_1 \) and \( a_2 \) are:

• A. 3, 3
• B. 3, 4
• C. 4, 3
• D. 4, 4

Hint:

Remember Vieta's formulas for a quadratic equation \( ax^2+bx+c=0 \). The sum of roots is \( -b/a \) and the product of roots is \( c/a \). For the characteristic equation \( m^2+a_1m+a_2=0 \), the sum of roots is \( -a_1 \) and the product is \( a_2 \). This provides a direct way to find the coefficients from the roots.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This question connects the solution of a second-order linear homogeneous differential equation with constant coefficients to the coefficients of the equation itself. The form of the solution tells us about the roots of the characteristic (or auxiliary) equation.

Step 2: Key Formula or Approach:
The differential equation is \( y'' + a_1 y' + a_2 y = 0 \). The corresponding characteristic equation is: \[ m^2 + a_1 m + a_2 = 0 \] The general solution is given as \( y = b_1 e^{-x} + b_2 e^{-3x} \). This form of solution arises when the characteristic equation has two distinct real roots. The roots are the coefficients of \(x\) in the exponents, which are \( m_1 = -1 \) and \( m_2 = -3 \).

Step 3: Detailed Explanation:
Since the roots of the characteristic equation \( m^2 + a_1 m + a_2 = 0 \) are \( -1 \) and \( -3 \), we can reconstruct the equation from its roots. A quadratic equation with roots \( r_1 \) and \( r_2 \) can be written as \( (m - r_1)(m - r_2) = 0 \), which expands to \( m^2 - (r_1+r_2)m + r_1r_2 = 0 \). Here, \( r_1 = -1 \) and \( r_2 = -3 \). - Sum of the roots: \( r_1 + r_2 = -1 + (-3) = -4 \) - Product of the roots: \( r_1 r_2 = (-1)(-3) = 3 \) The characteristic equation is: \[ m^2 - (-4)m + (3) = 0 \] \[ m^2 + 4m + 3 = 0 \] By comparing this with the general characteristic equation \( m^2 + a_1 m + a_2 = 0 \), we can identify the coefficients: - \( a_1 = 4 \) - \( a_2 = 3 \)
Step 4: Final Answer:
The respective values of \( a_1 \) and \( a_2 \) are 4 and 3.