Question

\[ e^{-x}\left(c_1\cos\sqrt{3}\,x+c_2\sin\sqrt{3}\,x\right)+c_3e^{2x} \] is the general solution of

• A. \(\dfrac{d^3y}{dx^3}+4y=0\)
• B. \(\dfrac{d^3y}{dx^3}-8y=0\)
• C. \(\dfrac{d^3y}{dx^3}+8y=0\)
• D. \(\dfrac{d^3y}{dx^3}-2\dfrac{d^2y}{dx^2}+\dfrac{dy}{dx}-2y=0\)

Hint:

From \(e^{\alpha x}\cos\beta x\) and \(e^{\alpha x}\sin\beta x\), write roots \(\alpha\pm i\beta\).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
We are given the general solution (which is the complementary function for a homogeneous DE) and need to find the corresponding differential equation. This is the reverse process of solving a DE. We identify the roots of the auxiliary equation from the solution, construct the auxiliary equation, and then form the differential equation.

Step 2: Key Formula or Approach:
1. From terms like \(e^{ax}(c_1\cos(bx) + c_2\sin(bx))\), we identify complex roots \(m = a \pm ib\). 2. From terms like \(c_k e^{rx}\), we identify a real root \(m = r\). 3. Combine the roots to form the auxiliary polynomial \((m-m_1)(m-m_2)\dots = 0\). 4. Convert the polynomial in \(m\) to a differential operator in \(D\).

Step 3: Detailed Explanation:
The given general solution is \(y = e^{-x}(c_1 \cos(\sqrt{3}x) + c_2 \sin(\sqrt{3}x)) + c_3e^{2x}\). Let's analyze the terms to find the auxiliary roots: - The term \(e^{-x}(c_1 \cos(\sqrt{3}x) + c_2 \sin(\sqrt{3}x))\) corresponds to complex conjugate roots of the form \(a \pm ib\), where \(a = -1\) and \(b = \sqrt{3}\). So, two roots are \(m = -1 \pm i\sqrt{3}\). - The term \(c_3e^{2x}\) corresponds to a real root \(m = 2\). So, the three roots of the auxiliary equation are \(m_1 = 2\), \(m_2 = -1 + i\sqrt{3}\), and \(m_3 = -1 - i\sqrt{3}\). The auxiliary equation is formed by the product of the factors corresponding to these roots: \[ (m - 2)(m - (-1 + i\sqrt{3}))(m - (-1 - i\sqrt{3})) = 0 \] \[ (m - 2)((m+1) - i\sqrt{3})((m+1) + i\sqrt{3}) = 0 \] Simplify the part with complex conjugates first: \[ (m - 2)((m+1)^2 - (i\sqrt{3})^2) = 0 \] \[ (m - 2)(m^2 + 2m + 1 - (-3)) = 0 \] \[ (m - 2)(m^2 + 2m + 4) = 0 \] Now, expand the full polynomial: \[ m(m^2 + 2m + 4) - 2(m^2 + 2m + 4) = 0 \] \[ m^3 + 2m^2 + 4m - 2m^2 - 4m - 8 = 0 \] \[ m^3 - 8 = 0 \] This is the auxiliary equation. The corresponding differential equation is obtained by replacing \(m^3\) with \(D^3y\) and the constant term with that constant times \(y\): \[ D^3y - 8y = 0 \quad \text{or} \quad \frac{d^3y}{dx^3} - 8y = 0 \]

Step 4: Final Answer:
The differential equation is \(\frac{d^3y}{dx^3} - 8y = 0\), which matches option (B).