Question

The solution of the differential equation \(\frac{d^3y}{dx^3} - 5\frac{d^2y}{dx^2} + 8\frac{dy}{dx} - 4y = 0\) is

• A. y \(=\) c\(_1\)e\(^x\) \(+\) (c\(_2\) \(+\) c\(_3\)x)e\(^{2x}\)
• B. y \(=\) c\(_1\)e\(^{2x}\) \(+\) (c\(_2\) \(+\) c\(_3\)x)e\(^x\)
• C. y \(=\) (c\(_1\) \(+\) c\(_2\)x \(+\) c\(_3\)x\(^2\))e\(^x\)
• D. y \(=\) (c\(_1\) \(+\) c\(_2\)x \(+\) c\(_3\)x\(^2\))e\(^{2x}\)

Hint:

Standard Auxiliary Root Rules:
- Real, distinct roots \( m_1, m_2 \rightarrow c_1 e^{m_1 x} + c_2 e^{m_2 x} \).
- Real, repeated roots \( m_1 = m_2 = m \rightarrow (c_1 + c_2 x)e^{mx} \).
- Complex conjugate roots \( \alpha \pm i\beta \rightarrow e^{\alpha x}(c_1 \cos \beta x + c_2 \sin \beta x) \).

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
This question asks for the general solution of a third-order homogeneous linear ordinary differential equation with constant coefficients.

Step 2: Key Formula or Approach:
For a linear homogeneous differential equation with constant coefficients, we find the general solution by solving its auxiliary (or characteristic) algebraic equation.
If \( m_1, m_2, m_3 \) are the roots of the auxiliary equation:
- Distinct real roots \( m_1 \neq m_2 \) yield solutions of the form \( e^{m_1 x} \).
- A repeated real root \( m \) of multiplicity 2 yields a solution of the form \( (c_1 + c_2 x)e^{mx} \).

Step 3: Detailed Explanation:

• Formulate the Auxiliary Equation:
- Replace \( \frac{d^n y}{dx^n} \) with \( m^n \):
\[ m^3 - 5m^2 + 8m - 4 = 0 \]

• Solve for the Roots of the Cubic Equation:
- Test simple integer roots using the rational root theorem. Let's test \( m = 1 \):
\[ (1)^3 - 5(1)^2 + 8(1) - 4 = 1 - 5 + 8 - 4 = 0 \] - Since \( m = 1 \) is a root, \( (m - 1) \) is a factor.
- Divide \( m^3 - 5m^2 + 8m - 4 \) by \( (m - 1) \) using synthetic division:
\[ m^3 - 5m^2 + 8m - 4 = (m - 1)(m^2 - 4m + 4) \] - Factor the remaining quadratic equation:
\[ m^2 - 4m + 4 = (m - 2)^2 \] - Thus, the roots of the auxiliary equation are:
\[ m = 1 \quad (\text{distinct}) \]
\[ m = 2, 2 \quad (\text{repeated with multiplicity 2}) \]

• Construct the General Solution:
- For root \( m = 1 \), the solution is \( c_1 e^x \).
- For repeated root \( m = 2 \) of multiplicity 2, the solution is \( (c_2 + c_3 x)e^{2x} \).
- Combine these linearly to find the general solution:
\[ y = c_1 e^x + (c_2 + c_3 x)e^{2x} \]


Step 4: Final Answer:
The general solution of the differential equation is \( y = c_1 e^x + (c_2 + c_3 x)e^{2x} \).
Therefore, the correct choice is option (A).