User:Egm6321.f12.team7.parikh.a/R3.10

=R*3.10=

Problem: Solve a L1-ODE-CC
Problem R*3.10 from lecture notes section 15.

Given: A L1-ODE-CC
Given a linear, first order, ordinary differential equation with constant coefficients of the form:

Find: The solution to the L1-ODE-CC
Show that the solution is given by:

using Euler's integrating factor method. Identify the integrating factor, homogeneous solution and particular solution. Also show that when the coefficients $$a$$ and $$b$$ vary with time, the solution becomes

Solution: Based on Euler's IFM

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On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions.
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($$) can be rearranged into the following form

where

Hence, the first exactness condition of N1-ODEs is satisfied. Thus, to satisfy the second exactness condition, the integrating factor, $$h$$, must be such that:

Assuming $$h_x (t,x) = 0$$ (i.e. $$h$$ is a function of $$t$$ only), the following condition must be satisfied, where $$n(t)$$ is a function of only $$t$$:

Substituting into ($$) from ($$) and ($$), produces

which indeed is a function of $$t$$ only. Solving for $$h$$ results in

where $$s$$ was used as a dummy integration variable. This $$h$$ is the integrating factor used to make the original ODE exact. Multiplying into both sides of the original ODE produces

From the product rule of differentiation, and observing that $$h_t = -ah$$, the left hand side of ($$) can be simplified into:

Integrating both sides from $$t_0$$ to $$t$$ and solving for x(t) produces

where $$\tau$$ is a dummy integrating variable and $$h(t_0) = 1$$. Substituting in ($$) for $$h$$ yields the solution given in ($$)

The first term on the right hand side is the homogeneous solution, as it is independent of the forcing function. The second term is the particular solution, as it depends on the nonhomogeneous term from the original ODE. When the coefficients $$a$$ and $$b$$ vary with time, the solution process is identical, except the integral on the right hand side of ($$) cannot be evaluated. Therefore, the integrating factor becomes:

Substituting into ($$) and noting that $$s$$ and $$\tau$$ are dummy integrating variables, the solution becomes