User:Egm6321.f12.team7.parikh.a/R4.3

=R*4.3=

Problem: First Integral for a class of exact L2-ODE-VC
Problem R*4.3 from lecture notes section 21.

Given: A class of exact L2-ODE-VC
Given a class of linear, 2nd order, ordinary differential equations with varying coefficients of the form:

Find: A general form of the first integral
Show that the first integral of the form given below generates a class of exact L2-ODE-VC of the form given in ($$).

where

Solution: Derive the first integral

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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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Integrating ($$) with respect to p yields:

The partial derivatives with respect to x and y are given by

Substituting back into the general form of L2-ODE-VC ($$)

where

Integrating ($$) results in

where

Substituting into ($$) produces

Taking the partial derivative with respect to y and comparing to Q(x) (since $$\phi_y = Q(x)$$ from ($$) and ($$))

Therefore, $$k_1 '$$ cannot be a function of y, and $$k_1$$ must be a constant. This means