Unit 7: Differential Equations

Executive Summary

Unit 7 models change with differential equations and finds solutions via slope fields, Euler's method, and separation of variables.

Modeling

"Rate proportional to the amount" translates to $\frac{dy}{dt} = ky$ — exponential growth ($k>0$) or decay ($k<0$). Solutions are $y = Ce^{kt}$.

Verifying solutions

To verify $y = f(x)$ solves a DE, differentiate and substitute into the equation; both sides must match.

Slope fields

A slope field draws short segments of slope $y' = f(x,y)$ at grid points. Match fields to equations by checking a fixed point's slope or where slopes are zero/vertical. Solution curves flow along the segments.

Euler's method

Step from the initial point: $y_{n+1} = y_n + h \cdot f(x_n, y_n)$ with step size $h$. Smaller steps improve accuracy; the method follows tangent lines.

Separation of variables

Rewrite as $g(y)\,dy = h(x)\,dx$, integrate both sides, and use the initial condition to find the particular solution. Watch for $\ln|y|$ algebra when solving for $y$.

Exam traps

Verify, don't solve, when asked to check a solution. Euler's method uses the CURRENT point's slope at every step. Don't forget the constant — then apply the initial condition to pin it down. Exponential models: the rate k multiplies t in the exponent.