Introduction
The general solution of a linear differential equation with constant coefficients is the sum of the complementary function (CF) and the particular integral (PI):
\[ y(x) = y_c(x) + y_p(x) \]
General Form & Operator Notation
We write the differential equation as:
\[ f(D)y = X(x), \quad D = \frac{d}{dx} \]
Complementary Function (CF)
Solve the homogeneous equation:
\[ f(D)y = 0 \]
The auxiliary equation \( f(m) = 0 \) gives roots that decide the form of CF.
Particular Integral (PI)
A particular solution is found as:
\[ y_p = \frac{1}{f(D)}X(x) \]
Steps to Solve
- Form the auxiliary equation and find CF.
- Apply rules for PI depending on \(X(x)\).
- Write general solution: \( y = y_c + y_p \).
Complementary Function (CF) — Roots of Auxiliary Equation
The Complementary Function (CF) depends on the nature of the roots of the auxiliary (characteristic) equation:
| Case | Roots of Auxiliary Equation | Form of Complementary Function \(y_c\) | Example |
|---|---|---|---|
| 1 | Two distinct real roots \(m_1, m_2\) | \(y_c = C_1 e^{m_1 x} + C_2 e^{m_2 x}\) | \(m^2-5m+6=0 \Rightarrow m=2,3 \Rightarrow y_c=C_1 e^{2x}+C_2 e^{3x}\) |
| 2 | \(n\) distinct real roots \(m_1,\dots,m_n\) | \(y_c = \displaystyle\sum_{k=1}^n C_k e^{m_k x}\) | \(m^3-6m^2+11m-6=0 \Rightarrow m=1,2,3 \Rightarrow y_c=\sum_{k=1}^3 C_k e^{m_k x}\) |
| 3 | One pair of complex conjugate roots \(\alpha \pm i\beta\) | \(y_c = e^{\alpha x}\big(C_1\cos\beta x + C_2\sin\beta x\big)\) | \(m^2+4m+13=0 \Rightarrow m=-2\pm3i \Rightarrow y_c=e^{-2x}(C_1\cos3x + C_2\sin3x)\) |
| 4 | Mixture of distinct real and (one or more) complex roots | Combine the corresponding terms: real roots give \(C e^{mx}\); complex pairs give \(e^{\alpha x}(C_1\cos\beta x+C_2\sin\beta x)\) | \(m=(1,\, -1,\, 2\pm i)\Rightarrow y_c=C_1e^{x}+C_2e^{-x}+e^{2x}(C_3\cos x + C_4\sin x)\) |
| 5 | Repeated real root of multiplicity \(r\) (e.g. \(m\) repeated) | \(y_c = (C_1 + C_2 x + C_3 x^2 + \dots + C_r x^{\,r-1}) e^{m x}\) | \(m^2-4m+4=0 \Rightarrow m=2,2 \Rightarrow y_c=(C_1+C_2x)e^{2x}\) |
| 6 | Repeated complex roots \((\alpha \pm i\beta)\) of multiplicity \(r\) | \(y_c = e^{\alpha x}\Big[(C_1 + C_2 x + \dots + C_r x^{\,r-1})\cos(\beta x) + (D_1 + D_2 x + \dots + D_r x^{\,r-1})\sin(\beta x)\Big]\) | \((m+2)^2+9=0 \Rightarrow m=-2\pm3i\) (repeated) \(y_c = e^{-2x}\big((C_1+C_2x)\cos3x+(D_1+D_2x)\sin3x\big)\) |
Worked Example
Solve \( (D^2 - 3D + 2)y = e^{2x} \):
Auxiliary equation: \( m^2 - 3m + 2 = 0 \implies m=1,2 \)
CF: \( y_c = C_1 e^x + C_2 e^{2x} \)
For PI: try \( y_p = \frac{e^{2x}}{f(2)} \). But \( f(2)=0 \), so multiply by \(x\).
\( y_p = x e^{2x} \)
General solution: \( y = C_1 e^x + C_2 e^{2x} + x e^{2x} \)
Tips & Pitfalls
- Check multiplicity of roots to adjust CF form.
- If denominator vanishes in PI, multiply trial by \(x\).
- Always verify by substitution.