# Definition Mathematical expression containing a function and one or more of its derivatives: $ \frac{d^ny(x)}{dx^n}+\dots+\frac{dy(x)}{dx}+y(x)=f(x) $ # First-Order Linear Differential Equations A ***first-order*** differential equation contains only first derivatives of the function A ***linear*** differential equation can be written as a sum of multiples of the function and its derivatives A differential equation with scalar multipliers has ***constant coefficients*** $ b_n\frac{d^ny(x)}{dx^n}+\dots+b_1\frac{dy(x)}{dx}+b_0y(x)=f(x) $ A differential equation whose *forcing function*, or right hand side, is equal to zero, is called ***homogeneous*** $ b_n\frac{d^ny(x)}{dx^n}+\dots+b_1\frac{dy(x)}{dx}+b_0y(x)=0 $ ## Solving $ y'+ay=0 $ $ y=Ce^{-at} $ ## Example $ \begin{align} \frac{dy}{dt}-5y&=0\\ y(0)&=3 \end{align} $ ## Solution $ y=Ce^{-5t} $ $ 3=Ce^{(0)t} $ $ C=3 $ $ \boxed{y=3e^{-5t}} $ # Second-Order Linear Differential Equations A ***second-order*** differential equation contains both first *and* second derivatives of the function $ y''+ay'+by=0 $ Second-order differential equations often describe real-world systems such as electrical circuits, spring-mass systems, fluid flows, heat transfer, etc. The behavior of these systems is classified by the amount of ***damping*** (electrical resistance, mechanical friction, pressure drop, thermal insulation, etc.) in the system: - underdamped - transient behavior gradually decays in an oscillatory manner to steady-state behavior - ***damping ratio*** $\zeta < 1$ - characteristic equation has two complex roots - overdamped - transient behavior decays sluggishly without oscillation to steady-state behavior - $\zeta>1$ - characteristic equation has two distinct real roots - critically damped - transient behavior reaches steady-state the quickest out of the three cases, without oscillation - $\zeta=1$ - characteristic equation has two identical real roots ## Solving Given the differential equation: $ y''+ay'+by=0 $ rewrite as the characteristic equation: $ r^2+ar+b=0 $ and find its roots $ r_{1,2}=\frac{-a\pm\sqrt{a^2-4b}}{2} $ - $a^2>4b$ - roots are real and different - system is overdamped - $a^2=4b$ - roots are real and identical - system is critically damped - $a^2<4b$ - roots are imaginary and take the form $(\alpha+i\beta)$ and $(\alpha-i\beta)$ Solution can appear as: $ y=C_1e^{r_1x}+C_2e^{r_2x} $ $ y=(C_1+C_2x)e^{rx} $ $ y=e^{\alpha x}(C_1\cos\beta x+C_2\sin\alpha x) $ $ \alpha = -a/2\qquad\beta=\frac{\sqrt{4b-a^2}}{2} $ ## Example What is the general solution to this homogeneous differential equation? $ y''-8y'+16y=0 $ ## Solution $ r^2-8r+16=0 $ $ r_{1,2}=\frac{8\pm\sqrt{8^2-4(16)}}{2}=4 $ System is critically damped $ \boxed{y=(C_1+C_2x)e^{4x}} $ # Nonhomogeneous Differential Equations $ y(x)=y_h(x)+y_p(x) $ - $y_h(x)$ - complementary (homogeneous) solution - $y_p(x)$ - particular solution ## Method of Undetermined Coefficients A method of finding the particular solution - can only be used if the forcing function (right hand side) is one of the following | form of $f(x)$ | form of $y_p(x)$ | | --------------------------------- | --------------------------------- | | $A$ | $B$ | | $Ae^{\alpha x}$ | $Be^{\alpha x}$ | | $A_1\sin\omega x+A_2\cos\omega x$ | $B_1\sin\omega x+B_2\cos\omega x$ | - $A_i$, $B_i$ not known --> ***undetermined coefficients***