Linear Differential Equation with
Constant Co-efficients

\(a \frac{d^2y}{dx^2} + b \frac{dy}{dx} + c = f(x) \)  ................... (i)

(a, b, c ∈ R and f:R → R is continuous)

Vocabulary

• Solution: --Select-- Any set Any polynomial Any function which satisfies Equation (i) is called solution. Submit

• Complementary Function (CF): --Select-- All the solutions All the Co-efficients Right Hand Side of the homogeneous equation \(a \frac{d^2y}{dx^2} + b \frac{dy}{dx} + c = 0 \) are collectively called CF. Submit

• Particular Integral (PI): --Select-- A specific solution A polynomial A set that satisfies Equation (i) is called PI. Submit

• Complete Integral (CI) of Differential Equation (i):

• CI = --Select-- CF PI CF + PI Submit

• --Select-- A specific solution is Partial solution is All solutions are represented by CI. Submit
  

• General solution: An expression that represents --Select-- all solutions partial solution specific solution is known as general solution. Submit

• Special Case f(x)=0:

  • CI = --Select-- CF PI Submit

  • --Select-- CF PI provides all solutions of Equation (i). Submit

Understanding

• Notation: T ≡ \(a D^2+ b D + c1 \), where D is the standard \( \frac{d}{dx} \) operator and 1 is the standard constant operator "one".


• T(y)=f(x) is the linear map equation corresponding to the equation --Select-- 𝑑𝑦 ⁄ 𝑑𝑥 = 𝑓(𝑥) 𝑑²𝑦 ⁄ 𝑑𝑥² = 𝑓(𝑥) given by (i) . Submit
• Solution of T(y)=f(x) is ker T + y_o, where --Select-- 𝑇(𝑦₀) = 𝑓(𝑥) 𝑦₀ = 𝑓(𝑥) 𝐷𝑦₀ = 𝑓(𝑥) . Submit
• Connection between T(y)=f(x) and Equation (i):
  
  
  • Complementary Function (CF) for Equation (i) corresponds to --Select-- ker 𝑇 + 𝑦₀ ker 𝑇 𝑦₀ . Submit
  • Particular Integral (PI) for Equation (i) corresponds to --Select-- ker 𝑇 + 𝑦₀ 𝑦₀ ker 𝑇 . Submit
  • Solution of linear map equation T(y)=f(x) is same as --Select-- complete integral particular integral complementary Function of Equation (i). Submit