Differential Equations
What you must understand
A differential equation relates a function to its derivatives. A solution satisfies the equation; the general solution includes a constant determined by an initial condition (initial value problem).
Separable equations: isolate variables so $g(y),dy=f(x),dx$, integrate both sides, then apply the initial condition. Recognize slope fields: a slope field shows $\frac{dy}{dx}$ at grid points; sketch solution curves by following segment slopes.
Exponential models $\frac{dy}{dt}=ky$ have solutions $y=Ce^{kt}$, used for growth/decay. Logistic models $\frac{dy}{dt}=ky(1-y/L)$ have a carrying capacity $L$. Interpret qualitative behavior (equilibria, where solutions increase/decrease) from the differential equation itself without solving.
Traps
- Forgetting $+C$ before applying the initial condition.
- Losing solutions when dividing by a variable expression.
- Misreading slope fields (check axis orientation and scale).
- Confusing growth ($k>0$) and decay ($k<0$) signs.
Worked example
Solve $\frac{dy}{dx}=2xy$ with $y(0)=3$. Separate: $\frac{1}{y}dy=2x,dx$. Integrate: $\ln|y|=x^2+C$, so $y=Ae^{x^2}$. With $y(0)=3$, $A=3$: $y=3e^{x^2}$.
Rule of thumb: separate, integrate both sides, then use the initial condition to pin down the constant; check slope fields by testing a sample point.