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Saturday, February 8, 2020

Integration factor method to solve linear differential equation

solve first order linear differential equation with Integration factor method

dydx+p(x)y(x)=q(x)
let v(x)=p(x)dx
The integration factor is I=ev(x).
d[ev(x)y(x)]dx=ev(x)[dydx+p(x)y(x)]=ev(x)q(x)
The solution of the differential equation is
y(x)=ev(x)[ev(x)q(x)dx+C]

[I(x)y]=I(x)q(x)
xx0[I(x)y]dx=I(x)y(x)I(x0)y(x0)=xx0I(x)q(x)dx
y(x)=I(x)1[I(x0)y(x0)+xx0I(x)q(x)dx]
Reference:
http://mathworld.wolfram.com/IntegratingFactor.html
https://www.stewartcalculus.com/data/CALCULUS%20Concepts%20and%20Contexts/upfiles/3c3-LinearDiffEqns_Stu.pdf

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