Initial Value Problem Non Homogeneous ODE |GYMAT101 Group B&C|S1 module2 |MAT102 S2 Module 3| Part8

Introduction

This tutorial provides a step-by-step guide to solving initial value problems for non-homogeneous ordinary differential equations (ODEs). Understanding these concepts is crucial for students in mathematics, especially in fields like electrical and physical sciences. We will break down the process into manageable steps, making it easier to grasp the methods involved.

Step 1: Understanding the Non-Homogeneous ODE

• A non-homogeneous ODE takes the form:

  [ y'' + p(x)y' + q(x)y = g(x) ]

  where:

  • ( y ) is the unknown function,
  • ( g(x) ) is a non-homogeneous term (i.e., not equal to zero).

• The general solution of a non-homogeneous ODE consists of two parts:

  • The complementary function (( y_c )): Solution of the associated homogeneous equation.
  • The particular solution (( y_p )): A specific solution to the non-homogeneous equation.

Step 2: Solve the Homogeneous Equation

1. Set the non-homogeneous term ( g(x) ) to zero:

   [ y'' + p(x)y' + q(x)y = 0 ]

2. Find the characteristic equation associated with this ODE:

   [ r^2 + p(r)r + q(r) = 0 ]

3. Solve for ( r ) to find the roots, which will help determine the form of the complementary function ( y_c ).

• Common cases:
  • Two distinct real roots: ( y_c = C_1e^{r_1x} + C_2e^{r_2x} )
  • Repeated roots: ( y_c = (C_1 + C_2x)e^{rx} )
  • Complex roots: ( y_c = e^{\alpha x}(C_1 \cos(\beta x) + C_2 \sin(\beta x)) )

Step 3: Find the Particular Solution

1. Choose a method to find ( y_p ):

   • Method of undetermined coefficients
   • Variation of parameters

2. For the method of undetermined coefficients, guess the form of ( y_p ) based on ( g(x) ):

   • Example guesses:
     • If ( g(x) = Ax + B ), then guess ( y_p = Cx + D ).
     • If ( g(x) = Ae^{kx} ), then guess ( y_p = Ce^{kx} ).

3. Substitute ( y_p ) back into the original non-homogeneous equation to solve for coefficients.

Step 4: Combine Solutions

• Combine the complementary function and the particular solution to get the general solution:

[ y = y_c + y_p ]

Step 5: Apply Initial Conditions

1. Use initial conditions provided in the problem (e.g., values of ( y ) and ( y' ) at a specific point) to solve for constants ( C_1 ) and ( C_2 ) in the general solution.
2. Substitute these constants back into the general solution to obtain the specific solution.

Conclusion

In this tutorial, we covered the process of solving initial value problems for non-homogeneous ODEs, including finding the complementary and particular solutions and applying initial conditions. To further solidify your understanding, practice solving various problems and refer to the provided playlists for additional resources on related topics.