A linear differential equation of the first order has the form |
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First we solve the corresponding homogeneous equation |
y' + P y = 0, dy/dx + P y = 0, dy/y = -P dx, log(y) = log(C) - ∫ P dx. |
The general solution of the equation is |
= C , = C v(x), v(x) = . |
The method called "variation of parameters" consists of finding a particular solution of the original equation in the form |
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C'(x) v(x) + C(x) (v'(x) + P v(x)) = Q. |
Since v(x) is a solution of the homogeneous equation |
C'(x) v(x) = Q, C'(x) = Q / v(x), C(x) = ∫ Q / v(x) dx. |
The general solution of the original equation is the sum of the general solution of the homogeneous equation and of the particular solution: |
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The method is valid even in case of an equation of the form |
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