Integrating factor: Difference between revisions
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(Created page with "'''Integrating factors''' is a method to solve scalar Linear First Order ODE<nowiki/>s. Consider <math>y' + p(t) y = g(t)</math> We multiply bothsides by integrating factor <math>\mu(t)</math> such that the left hand side to be the ''exact derivative of a product.'' For this to work, we need <math>\mu' = \mu p</math> Applying the integrating factor, we have <math>\begin{aligned} \mu(t) y + \mu(t) p(t) y &= \mu(t) g(t) \\ \mu y' + \mu ' y &= \mu g \\ (\mu y)' &=...") |
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<math>\mu' = \mu p</math> | <math>\mu' = \mu p</math> | ||
<math>\mu(t) = C \exp\left[ \int p dt \right]</math> | |||
Applying the integrating factor, we have | Applying the integrating factor, we have | ||
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From there, simple integration and algebra will solve the equation. | From there, simple integration and algebra will solve the equation. | ||
<math>y(t) = \frac{1}{\mu(t)}\int \mu g dt + C </math> |
Revision as of 21:52, 17 April 2024
Integrating factors is a method to solve scalar Linear First Order ODEs.
Consider
We multiply bothsides by integrating factor such that the left hand side to be the exact derivative of a product. For this to work, we need
Applying the integrating factor, we have
From there, simple integration and algebra will solve the equation.