Given Differential Equation :
\(\frac {dy}{dx} +3y = e^{-2x}\)
Formula :
i) \(\int\) 1dx = x
ii) \(\int e^{kx} dx = \frac {e^{kx}}{x}\)
iii) General solution :
For the differential equation in the form of
\(\frac{dy}{dx} +Py = Q\)
General solution is given by,
y. (I. F.) = \(\int\) Q. (I. F.) dx + c
Where, integrating factor,
I. F. = \(e^{\int p\,dx}\)
Given differential equation is
\(\frac {dy}{dx} +3y = e^{-2x} \) ……eq(1)
Equation (1) is of the form
\(\frac {dy}{dx} + Py = Q\)
Where, \(P= 3 \, and\, Q = e^{-2x}\)
Therefore, integrating factor is
General solution is
Dividing above equation by (e-3x),
Therefore general solution is
\(y = e^ {-2x} + ce^{-3x}\)
