Notes on Solving Differential Equations

Overview

• Focus on solving second-order homogeneous differential equations using complementary functions.
• Utilize auxiliary equations for factorization and simplify equations to find roots.
• Different types of auxiliary equations and corresponding complementary functions are discussed with examples.

Example 1: Solving $d^2y/dx^2 - 7dy/dx - 44y = 0$

1. Given Equation: $d^2y - 7dy - 44y = 0$
2. Auxiliary Equation: $m^2 - 7m - 44 = 0$
   • Factorization: $(m + 4)(m - 11) = 0$
   • Roots: $m_1 = -4$, $m_2 = 11$
   • Complementary Function: $C_1 e^{-4x} + C_2 e^{11x}$

Example 2: Solving $D^4 - 5D^2 + 4 = 0$

1. Auxiliary Equation: $m^4 - 5m^2 + 4 = 0$
   • Simplified to: $(m^2 - 4)(m^2 - 1) = 0$
   • Factorization: $(m - 2)(m + 2)(m - 1)(m + 1) = 0$
   • Roots: $m_1 = 2$, $m_2 = -2$, $m_3 = 1$, $m_4 = -1$
   • Complementary Function: $C_1 e^{2x} + C_2 e^{-2x} + C_3 e^{x} + C_4 e^{-x}$

Example 3: Solving $D^3 - 2D^2 - 4D + 8 = 0$

1. Auxiliary Equation: $m^3 - 2m^2 - 4m + 8 = 0$
   • Factorization: $(m - 2)^2(m + 2) = 0$
   • Roots: $m_1 = 2$ (double root), $m_2 = -2$
   • Complementary Function: $(C_1 + C_2 x)e^{2x} + C_3 e^{-2x}$

Example 4: Solving $D^4 - M^4 = 0$

1. Auxiliary Equation: $D^4 - M^4 = 0$
   • Factorization: $(D^2 - M^2)(D^2 + M^2) = 0$
   • Further Factorization: $(D - M)(D + M)(D - Mi)(D + Mi) = 0$
   • Roots: $m_1 = M$, $m_2 = -M$, $m_3 = iM$, $m_4 = -iM$
   • Complementary Function:
     • For real roots: $C_1 e^{Mx} + C_2 e^{-Mx}$
     • For complex roots: $C_3 \cos(Mx) + C_4 \sin(Mx)$

Example 5: Solving $D^4 + M^4 = 0$

1. Modification: Added and subtracted $2M^2D^2$
2. New Form: $(D^2 + M^2)^2 - (2MD)^2 = 0$
3. Auxiliary Equation: Formed using quadratic formula for complex roots.
   • Complementary Function involves $e^{\alpha x}(C_1 \cos(\beta x) + C_2 \sin(\beta x))$
   • Roots are complex, involving $\alpha$ and $\beta$ terms.

Key Concepts

• Auxiliary Equation: Crucial for determining roots which lead to the complementary function.
• Roots:
  • Real and distinct roots result in simple exponential terms in the solution.
  • Complex roots introduce trigonometric functions (sine and cosine) into the solution.
• Factorization: Essential for simplifying and finding solutions, often involving completing squares or recognizing patterns.

Conclusion

• Understanding the structure and manipulation of auxiliary equations is fundamental for solving differential equations and finding the complementary function.