: In first-order equations, the constant of integration is vital for applying initial conditions.
The Laplace Transform is a powerful tool that turns differential equations into algebraic equations, making them much easier to solve—especially when dealing with discontinuous forcing functions (like a switch turning on). Strategy: Transform the DE into the -domain, solve for , and then use an inverse transform table to find 5. Systems of Differential Equations differential equations lecture notes
Solve ( \frac{dy}{dx} + 2y = e^{-x} ). Integrating factor : ( \mu(x) = e^{\int 2,dx} = e^{2x} ). Multiply through: ( e^{2x}y' + 2e^{2x}y = e^{x} ) Left side is ( \frac{d}{dx}(e^{2x}y) = e^{x} ) Integrate: ( e^{2x}y = e^{x} + C ) Thus ( y = e^{-x} + Ce^{-2x} ). : In first-order equations, the constant of integration