When the explicit functions y = f(x) + cg(x) form the solution of an ODE, g is called the complementary function; f is the particular integral. Example of Solution Using a …

The complementary equation is \(y″−2y′+5y=0\), which has the general solution \(c_1e^x \cos 2x+c_2 e^x \sin 2x\) (step 1). Based on the form \(r(x)=10x^2−3x−3\), our initial guess for the …

Complete solution of equation is given by C.F + P.I. where C.F. denotes complimentary function and P.I. is particular integral. When , then solution of equation is given by C.F

Understand the concept of Complementary Function in solving linear differential equations. Learn methods of finding complementary functions with practical examples.

The complementary function is obtained by solving the equivalent homogeneous differential equation, i.e. with f(x) = 0, and the particular integral by considering the form of the f(x) …

The function that solves the homogeneous equation is known as the complementary function, yc y c. The general solution to the differential equation is given by adding the particular and …

The general solution of a linear differential equation is the sum of a particular integral and a complementary function. General solution = particular integral + complementary function …

We will use complementary functions and particular integrals to solve y′ + y = p(x); is constant. We know that the general solution is y(x) = f(x) |{z} particular integral (P.I.) + Cg(x) | {z } …

If the auxiliary equation has complex roots, α + β i and α − β i , then the complementary function is: y cf = e α x ( A cos β x + B sin β x )

Any solution, ~y_2, of the equation _ ~Q ( ~y_2 ) _ = _ ~f ( ~x ) _ is called a #~{particular integral} of the second order differential equation. The technique is therefore to find the complementary …