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 = y/x... wait: original DE: dy/dx = -y/x. Replace: dy/dx = 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^(x/k) (exponentials)
• Curve where tangent length = constant: y*sqrt(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 - (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.