Linear Differential Equation with Constant Co-efficients

\(a \frac{d^2y}{dx^2} + b \frac{dy}{dx} + c = f(x) \)  ................... (i)
where a, b, c ∈ R and f:R → R is continuous

Explaining Vocabulary

Solution: A function which satisfies the given equation

Complementary Function (CF): All the solutions of homogeneous equation \(a \frac{d^2y}{dx^2} + b \frac{dy}{dx} + c = 0 \)  .......... (ii)

Particular Integral (PI): A specific solution that satisfies eq. (ii)

Complete Integral (CI) of (i):
● Provides all solutions of eq. (i)
● CI = CF + PI

Special Case: f(x)=0
● CF provides all solutions of eq. (i), so that CI = CF

Understanding

• Notation: T ≡ \(a D^2+ b D + cI \)
• Linear map equation corresponding to (i): T(y)=f(x)   ........... (iii)
• Solution of (iii): ker T + y_o, where T(y_o) = f(x)
• Connection of (iii) with Linear Differential Equation (i):
  • Complementary Function (CF) for eq. (i) corresponds to ker T
  • Particular Integral (PI) for eq. (i) corresponds to y_o
  • Solution of linear map equation is same as complete integral of linear differential equation