Section 7.2 General Solutions
In any nonhomogeneous differential equation, the solution should, to some extent, resemble the right-hand side and we can logically guess what function the solution should look like. Yet, hidden within the overall solution could be terms that cancel out each other, affecting only the structure without contributing directly to the right-hand side.
To explore this, consider the following linear nonhomogeneous constant coefficient (LNCC) equation alongside its homogeneous counterpart:
Let
be the solution to the homogeneous equation
(33), and
be the solution to the nonhomogeneous equation
(32), so we have
Now, adding these equations together and rearranging the terms gives us
This shows that not only is
the solution to the nonhomogeneous equation
(32), but so is
This happens because the terms in
simplify to zero and the terms in
simplify to
From the previous section, we know
is a solution to the LNCC equation
(32) and
is the solution to the LHCC equation
(33). Therefore, the combined solution
is a solution to the LNCC equation
(32) where
contains the terms that simplify to zero, whereas,
contains the terms that simplify to
when
is substituted into
(32).
This example leads us to the following concept for the general solution of a LNCC differential equation.
LNCC General Solution Parts.
The linear nonhomogeneous constant coefficient (LNCC) equation
has a solution with a homogeneous and particular part given by
where found by solving the LHCC equation
Here,
contains the terms of the solution that cancel out, whereas,
contains the terms that simplify to
when
(35) is substituted into
(34).
Example 14.
Find the general solution to the differential equation
given that the the particular solution is known to be
Solution. Solution
The general solution has the form:
Since is given, we only need to solve the homogeneous equation:
Using the characteristic equation:
The homogeneous solution is:
Therefore, the general solution is:
Before we can tackle solving these equations from scratch, we need a strategy for finding the particular solution which will be covered in the upcoming sections.
Reading Questions Check-Point Questions
1. If and what is the general solution to the LNCC equation?
- Correct! The general solution combines the homogeneous and particular parts.
- Incorrect. The signs in the particular solution are wrong.
- Incorrect. This is only the homogeneous solution, not the complete general solution.
- Incorrect. This is only the particular solution, not the full general solution.
2. What is the purpose of the particular solution in solving a non-homogeneous linear differential equation?
What is the purpose of the particular solution in solving a non-homogeneous linear differential equation?- To represent the general solution of the homogeneous equation.
- Incorrect. The general solution of the homogeneous equation is called the complementary solution, not the particular solution.
- To determine the coefficients of the characteristic equation.
- Incorrect. The characteristic equation is related to the complementary solution and does not involve the particular solution.
- To account for the non-homogeneous term on the right-hand side of the equation.
- Correct! The particular solution is chosen to match the form of and account for its influence in the equation.
- To simplify the process of solving the differential equation.
- Incorrect. The particular solution addresses the specific form of and is part of solving the non-homogeneous equation, but its purpose isn’t simplification.
3. How is the homogeneous solution of an LNCC equation typically found?
- By solving the characteristic equation associated with the homogeneous equation.
- Correct! The characteristic equation provides the exponents for the homogeneous solution.
- By integrating the equation twice.
- Incorrect. Solving the characteristic equation is the standard method for homogeneous solutions.
- By guessing the solution and checking.
- Incorrect. The characteristic equation is the systematic way to find the homogeneous solution.
- By using boundary conditions.
- Incorrect. Boundary conditions are used to find specific constants, not to find
4. Which of the following statements are true about the particular part of the solution, of an LNCC equation?
5. Which of the following statements are true about the homogeneous part of the solution, of an LNCC equation?
You have attempted
1 of
6 activities on this page.