User:Egm6321.f12.team7.parikh.a/R3.1

=R*3.1=

Problem: Show Redundancy of Integration Constants
Problem R*3.1 from lecture notes section 12.

Given: L1-ODE-VC
Given a general non-homogenous, linear, first order differential equation with variable coefficients of the form:

Find: Redundancy of Integration Constants
Show how only one integration constant is needed.

Solution: Solve using Euler IFM with only one constant

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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 written as:

where

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

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

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

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

where $$s$$ is a dummy variable for integration. From the original ODE in ($$)

From ($$)

Substituting into ($$)

By definition of the product rule of differentiation

where q is a dummy variable for integration. Substituting ($$) into the solution yields

Since $$k_2$$ and $$C$$ are constants, the term $$k_2 / C$$ can be replaced with one constant, $$C_2$$. Hence, only one integration constant is necessary in the solution of the linear, first order, ordinary differential equation with variable coefficients of the form given by ($$). Intuitively, this makes sense since the differential equation is first order, so there should only be one integration constant.

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