Differential Equations
Core ideas
A differential equation relates a function to its derivatives. A solution is a function; an initial condition pins down a particular solution. Slope fields visualize $\frac{dy}{dx}=f(x,y)$: each point gets a small segment with slope $f(x,y)$, so you can sketch solution curves qualitatively.
Separation of variables works when the equation factors as $g(x)h(y)$: separate, integrate both sides, and solve for $y$ if possible. Exponential growth and decay models $\frac{dy}{dt}=ky$ with solution $y=Ce^{kt}$; logistic growth models bounded growth with a carrying capacity. Euler's method approximates solutions numerically using tangent-line steps.
Traps
- Forgetting the constant of integration when separating variables; it becomes the constant in the particular solution.
- Applying separation of variables when the equation does not factor into a product of a function of $x$ and a function of $y$.
- Misreading a slope field: slopes depend on both coordinates, not just one.
- In exponential models, the sign of $k$ determines growth versus decay.
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 $y=Ae^{x^2}$. Using $y(0)=3$ gives $A=3$, hence $y=3e^{x^2}$.