Growth and Decay: dN/dt = kN has solution N = $N_0$ e^(kt). k > 0: growth, k < 0: decay. Half-life: T = ln 2/|k|. Doubling time: T = ln 2/k. These problems appear as: bacterial growth, radioactive decay, population models.

Newton's Law of Cooling: dT/dt = -k(T - $T_s$) where $T_s$ is surrounding temperature. Solution: T = $T_s$ + ($T_0$ - $T_s$)e^(-kt). Common JEE format: given T at two times, find T at a third time.

Orthogonal Trajectories: Given family F(x,y,c) = 0: (1) Find DE by eliminating c, (2) Replace dy/dx by -dx/dy, (3) Solve the new DE. Example: xy = c gives y + xy' = 0, OT: dy/dx = $\frac{y}{x}$... wait: original DE: dy/dx = -y/x. Replace: dy/dx = $\frac{x}{y}$. Solving: $y^2$ - $x^2$ = C (hyperbolas).

Geometric Applications:

• Curve where subnormal = constant: y*y' = k, giving $y^2$ = 2kx + C (parabolas)
• Curve where subtangent = constant: y/y' = k, giving y = Ce^$\frac{x}{k}$ (exponentials)
• Curve where tangent length = constant: y*sqrt$\frac{1+y'^2}{y}$' = k

Mixture Problems: A tank has V liters with S kg dissolved substance. Inflow: $c_{in}$ kg/L at $r_{in}$ L/min. Outflow: concentration S/V at $r_{out}$ L/min. DE: dS/dt = $c_{in}$ * $r_{in}$ - $\frac{S}{V}$*$r_{out}$. If $r_{in}$ = $r_{out}$: V is constant, giving a linear DE. If $r_{in}$ != $r_{out}$: V changes with time.