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The image presents a question regarding the complete solution of the differential equation \((D - m_1 D' - a_1)(D - m_2 D' - a_2) = 0\), where \(m_1 \neq m_2\). The answer options are:

- (a) \( z = e^{sx} f_1(y + m_1x) + e^{sx} f_2(y + m_1x) + e^{sx} f_3(y + m_1x) \dots \)
- (b) \( z = e^{sx} f_1(y + m_1x) + e^{sx} f_2(y + m_1x) \)
- (c) \( z = e^{sx} f_1(y + m_1x) + x e^{sx} f_2(y + m_1x) \)
- (d) \( z = e^{sx} f_1(y + m_1x) + x e^{sx} f_2(y + m_1x) + x^2 e^{sx} f_3(y + m_1x) \)

This is a question on solving linear partial differential equations using the method of characteristic equations, typically arising in the context of second-order PDEs with constant coefficients.

The complete solution for such an equation depends on the nature of the roots of the auxiliary equation. Since \(m_1 \neq m_2\), we expect two distinct roots. In such cases, the general solution typically includes terms of the form \(e^{sx}\), with each characteristic root leading to functions \(f_i\) depending on \(y + m_ix\). Additionally, higher powers of \(x\) (as seen in options c and d) indicate solutions where the roots of the characteristic equation are repeated or have certain multiplicities.

Here, option **(d)** is correct, as it contains the complete solution with terms that account for distinct roots with corresponding powers of \(x\), which is typical for such equations when \(m_1 \neq m_2\).

Would you like to explore the reasoning in more detail, or discuss any related concepts? Here are some related questions:

1. How are characteristic equations derived from second-order partial differential equations?
2. What role do roots of the characteristic equation play in the solution of PDEs?
3. Can you explain the method of undetermined coefficients used in solving PDEs?
4. What happens when the roots of the characteristic equation are repeated?
5. How do we handle non-homogeneous terms in such differential equations?

**Tip:** Understanding the characteristic equation is key to solving linear partial differential equations, as it determines the form of the general solution!

What is the complete solution of (D - m1 D' - a1)(D - m2 D' - a2) = 0, where m1 ≠ m2?

Learn how to find the complete solution for a differential equation of the form (D - m1 D' - a1)(D - m2 D' - a2) = 0 with distinct roots (m1 ≠ m2). Understand the method of characteristics and the role of exponential and polynomial terms in the solution.

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