Education • All: 2.2: First Order Linear Equations

FIRST ORDER LINEAR EQUATIONS

Homogeneous linear first order differential equation. Integrating via an integrating factor.

1.Standard equation:

$\large \frac{dy}{dx}+P(x)y=g(x)$

2.Obtaining integrating factor:

$\large \mu =e^{\int P(x)dx}$

3. Multiply integrating factor to all terms in the equation.
(Note: middle term will always go to zero)

$\large \mu \frac{dy}{dx}\;\;{\color{Red}\underset{goes\;to\;zero}{-2\mu y} }\;\;=\mu0\\$

Example (1):

$\large \\\frac{dx}{dy}-2y=0\;\;\;(1.Given\;equation) \\\\\mu =e^{\int -2dx}=e^{-2x}\;\;\;(2.Find\;integrating\;factor) \\\\e^{-2x}\frac{dy}{dx}=0\times e^{-2x}\;\;\;(3.Multiply\;integrating\;factor) \\\\\int e^{-2x}dy=\int 0dx\;\;\;(4.Integrate\;both\;sides) \\\\e^{-2x}y=0+C\;\;\;(5.Solve\;for\;y) \\\\y=Ce^{2x}$

Example (2):

$\large \\3\frac{dy}{dx}+12y=4\;\;\;\;\;(1.Given\;equation) \\\\\frac{dy}{dx}+4y=\frac{4}{3}\;\;\;\;\;(2.Write\;equation\;in\;standard\;form) \\\\\mu =e^{\int4dx}=e^{4x}\;\;\;\;\;(3.Find\;integrating\;factor) \\\\e^{4x}\frac{dy}{dx}=\frac{4}{3}e^{4x}\;\;\;\;\;(4.Multiply\;integrating\;factor) \\\\\int e^{4x}dy=\int \frac{4}{3}e^{4x}\;\;\;\;\;(5.Integrate\;both\;sides) \\\\e^{4x}y=\frac{1}{3}e^{4x}+C\;\;\;\;\;(6.Solve\;for\;y) \\\\y=\frac{1}{3}+Ce^{-4x}$

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2.2: First Order Linear Equations

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