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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:
(32)y4y+3y= 9x(33)y4y+3y= 0
Let yh be the solution to the homogeneous equation (33), and yp be the solution to the nonhomogeneous equation (32), so we have
yp4yp+3yp= 9xyh4yh+3yh= 0
Now, adding these equations together and rearranging the terms gives us
yh4yh+3yh+yp4yp+3yp= 9xyh+yp4yh+4yp+3yh+3yp= 9x(yh+yp)4(yh+yp)+3(yh+yp)= 9x
This shows that not only is yp the solution to the nonhomogeneous equation (32), but so is yh+yp. This happens because the terms in yh simplify to zero and the terms in yp simplify to 9x.
From the previous section, we know yp=3x+4 is a solution to the LNCC equation (32) and yh=c1ex+c2e3x is the solution to the LHCC equation (33). Therefore, the combined solution
y=yh+yp=c1ex+c2e3xsimplify to 0+3x+4simplify to 9x
is a solution to the LNCC equation (32) where yh contains the terms that simplify to zero, whereas, yp contains the terms that simplify to f(x) when y 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
(34)any(n)++a1y+a0y=f(x)
has a solution with a homogeneous and particular part given by
(35)y=yh+yp.
where yh, found by solving the LHCC equation
any(n)++a2y+a1y+a0y=0.
Here, yh contains the terms of the solution that cancel out, whereas, yp contains the terms that simplify to f(x) when (35) is substituted into (34).

Example 14.

Find the general solution to the differential equation
y4y12y=3e5t
given that the the particular solution is known to be yp(t)=37e5t.
Solution. Solution
The general solution has the form:
y=yh+yp
Since yp is given, we only need to solve the homogeneous equation:
y4y12y=0
Using the characteristic equation:
r24r12= 0(r6)(r+2)= 0r1=6,r2=2
The homogeneous solution is:
yh(t)=c1e2t+c2e6t
Therefore, the general solution is:
y=c1e2t+c2e6t37e5t
Before we can tackle solving these equations from scratch, we need a strategy for finding the particular solution yp, which will be covered in the upcoming sections.

Reading Questions Check-Point Questions

1. If yh=c1ex+c2e2x and yp=5x3, what is the general solution to the LNCC equation?

  • y=c1ex+c2e2x+5x3.
  • Correct! The general solution combines the homogeneous and particular parts.
  • y=c1ex+c2e2x5x+3.
  • Incorrect. The signs in the particular solution are wrong.
  • y=c1ex+c2e2x.
  • Incorrect. This is only the homogeneous solution, not the complete general solution.
  • y=5x3.
  • Incorrect. This is only the particular solution, not the full general solution.

2. What is the purpose of the particular solution yp in solving a non-homogeneous linear differential equation?

    What is the purpose of the particular solution yp 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 f(x) on the right-hand side of the equation.
  • Correct! The particular solution is chosen to match the form of f(x) 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 f(x) and is part of solving the non-homogeneous equation, but its purpose isn’t simplification.

3. How is the homogeneous solution yh 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 yh.

4. Which of the following statements are true about the particular part of the solution, yp, of an LNCC equation?

    Which of the following statements are true about the particular part of the solution, yp, of the LNCC equation
    (36)any(n)++a1y+a0y=f(x)
  • It is part of the general solution of (36).
  • Correct! The general solution is the sum of the homogeneous solution yh and the particular solution yp.
  • It contains the terms in the general solution that simplify to zero when you plug it into the left-side of (36).
  • Incorrect. yh is the solution to the homogeneous part, while yp is the particular solution.
  • It contains the terms in the general solution that account for f.
  • Correct. The terms of yp simplify to f when plugged into (36).
  • It is found by solving an LHCC equation.
  • Incorrect. yh is the solution to the homogeneous equation, and yp solves the nonhomogeneous equation.
  • It contains constants of integration.
  • Incorrect The homogeneous solution includes constants of integration that are determined by the initial conditions.
  • It resembles f.
  • Correct! The particular solution looks like the non-zero right-hand side of the equation.

5. Which of the following statements are true about the homogeneous part of the solution, yh, of an LNCC equation?

    Which of the following statements are true about the homogeneous part of the solution,, yp, of the LNCC equation
    (37)any(n)++a1y+a0y=f(x)
  • It is part of the general solution of (37)
  • Correct! The general solution is the sum of the homogeneous solution yh and the particular solution yp.
  • It contains the terms in the general solution that simplify to zero when you plug it into the left-side of (37).
  • Correct! yh is the solution to the homogeneous part, while yp is the particular solution.
  • It contains the terms in the general solution that account for f.
  • Incorrect. yh is related to the homogeneous equation, while the terms of yp simplify to f when plugged into (37).
  • It is found by solving an LHCC equation.
  • Correct! yh is the solution to the homogeneous equation, and yp solves the nonhomogeneous equation.
  • It contains constants of integration.
  • Correct! The homogeneous solution includes constants of integration that are determined by the initial conditions.
  • It resembles f.
  • Incorrect The particular solution accounts for the non-zero right-hand side of the equation.
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