When linear differential equation can also directly be solved by variable separable method? Consider the linear differential equation (dy)/(dx)+p(x)y=q(x). 1. What conditions may be imposed on functions p(x) and/or q(x) so that it could directly be solved by variable separable method rather than integrating factor e^(int p(x) dx). 2. In the case if it can be solved by both variable separable and integrating factor method, which would be more suitable?

Adison Rogers

Adison Rogers

Answered question

2022-11-02

When linear differential equation can also directly be solved by variable separable method?
Consider the linear differential equation d y d x + p ( x ) y = q ( x ) . 1. What conditions may be imposed on functions p ( x ) and/or q ( x ) so that it could directly be solved by variable separable method rather than integrating factor e p ( x ) d x . 2. In the case if it can be solved by both variable separable and integrating factor method, which would be more suitable?

Answer & Explanation

Regan Holloway

Regan Holloway

Beginner2022-11-03Added 17 answers

Comparing the general linear ODE and the separable linear ODE,
d y d x + p ( x ) y = q ( x ) and d y d x = f ( x ) ( a y + b ) ,
we note that the linear ODE is separable if and only if
q ( x ) p ( x ) y = f ( x ) ( a y + b ) ,
with a new function f. This is the case if q ( x ) = c p ( x ) for some constant c R , since then we obtain q ( x ) p ( x ) y = p ( x ) ( c y )
via separation of variables or via an integrating factor is a matter of preference, and with both methods we obtain the general solution
y ( x ) = c + C e P ( x ) , C R ,
where P is an antiderivative of p.

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