Differential Equations
Differential Equations
UPMSP checks whether you can classify an equation and choose a solution method. The order is the highest derivative present; the degree is the power of the highest derivative after the equation is made polynomial in derivatives. A general solution contains as many arbitrary constants as the order, while a particular solution uses given conditions.
Core rules
- Variable separable: write dy/dx=f(x)g(y), separate, and integrate both sides.
- Homogeneous equations: substitute y=vx, so dy/dx=v+x dv/dx.
- Linear first-order: dy/dx+Py=Q, with integrating factor e^{∫P dx}; then d/dx(y·IF)=Q·IF.
- After integration, include C and apply initial conditions only after integrating.
- Check whether a constant solution was divided away during separation.
Worked example
Solve dy/dx=xy. Separate: dy/y=x dx, assuming y≠0. Integrate: ln|y|=x^2/2+C. Exponentiate: |y|=e^C e^{x^2/2}, so y=K e^{x^2/2} with K≠0. The separated solution y=0 also satisfies the original equation; it corresponds to K=0, so the general solution is y=Ce^{x^2/2}.
Traps
Confusing order and degree; losing the equilibrium solution y=0; adding C before integrating both sides correctly; using the integrating factor without multiplying Q; applying an initial condition to the derivative instead of the solution.