List of formulas and basic concepts related to linear inhomogeneous differential equations of the first order:
- The general equation of a linear inhomogeneous differential equation of the first order:
[
y’ + P(x)y = Q(x)
]
where ( P(x) ) and ( Q(x) ) are given functions. - Integrating factor:
[
\mu(x) = e^{\int P(x) \, dx}
]
Used to simplify the equation. - Multiplying the equation by an integrating multiplier:
[
\mu(x)(y’ + P(x)y) = \mu(x)Q(x
) - Transformation of the equation:
[
\frac{d}{dx}[\mu(x)y] = \mu(x)Q(x)
] - Integration of both sides:
[
\mu(x)y = \int \mu(x)Q(x) \, dx + C
]
where ( C ) is an arbitrary constant. - Solution for ( y ):
[
y = \frac{1}{\mu(x)}\left(\int \mu(x)Q(x) \, dx + C\right)
]
These formulas will help you solve linear nonhomogeneous first-order differential equations