6. Definition
Variation of parameters is a method used to find particular solutions to non-
homogeneous linear differential equations. It is a general method for finding
a particular solution of a differential equation by replacing the constants in
the solution of a related equation by functions and determining these
functions so that the original differential equation will be satisfied.
When is it used?
In cases where the method of undetermined coefficients fails, particularly
when the non-homogeneous term is a combination of functions.
8. Linear Differential Equations
A linear equation or polynomial, with one or more terms, consisting of the derivatives of the
dependent variable with respect to one or more independent variables is known as a linear
differential equation.
A general first-order differential equation is given by the expression:
ππ¦
ππ₯
+ ππ¦ = π where y is a function and dy/dx is a derivative.
The solution of the linear differential equation produces the value of variable y.
Non-Homogeneous Equations
A nonhomogeneous linear differential equation is an ordinary differential equation (ODE) in which
the dependent variable and its derivatives are present on one side of the equation, and there is a
non-zero function on the other side. The general form of a nonhomogeneous linear differential
equation of the second order is:
yβ²β²(x)+p(x)yβ²(x)+q(x)y(x)=r(x) , where p and q are co-efficients
We can find both the Complementary Function (CF) and the Particular Integral (PI) of the equation.
10. Steps in Variation of Parameters
1.Find the Complementary Solution (Homogeneous
Part): Solve the associated homogeneous differential
equation.
2.Find the Particular Solution (Particular Part): Assume
a particular solution and determine its undetermined
coefficients.
3.Combine Solutions: Form the general solution by
combining the complementary and particular solutions.
16. Pros and Cons of Variation of
Parameters
β’Advantages: Flexibility in handling various
non-homogeneous terms.
β’Limitations: Complexity may increase
with the nature of the differential equation.