Q1. Find the order and degree of d²y/dx² + (dy/dx)³ = 0.
Highest derivative is d²y/dx² (a second derivative) ⇒ order = 2
That highest-order derivative appears to the power 1 ⇒ degree = 1
Differential Equations — Class 12
A differential equation links a function to its own rate of change. Learn to classify one and solve the simplest kind.
OrderThe order of the highest derivative appearing in the equation.
DegreeThe power of the highest-order derivative, once the equation is a polynomial in its derivatives.
General & particular solutionThe general solution has arbitrary constants; a particular solution fixes them using given conditions.
Variable-separable methodIf dy/dx = f(x)·g(y), separate as dy/g(y) = f(x)dx and integrate both sides.
Solved examples
Q2. Solve dy/dx = y (variable separable).
Separate: dy/y = dx
Integrate both sides: ln|y| = x + C
So y = Ae^x (where A = e^C is a new constant)
Formula sheet
| Quantity | Formula |
|---|---|
| Variable separable | ∫dy/g(y) = ∫f(x)dx |
| Order example | d²y/dx² + y = 0 has order 2 |
Common mistakes & exam wins
- Order looks at which derivative is highest; degree looks at the POWER of that highest derivative.
- For variable-separable equations, get all y-terms (with dy) on one side and all x-terms (with dx) on the other before integrating.
- A "particular solution" always comes from applying an initial condition to the general solution.
Frequently asked questions
What is the order of a differential equation?
The order of the highest derivative that appears in the equation.
What is the variable-separable method?
A technique for solving dy/dx = f(x)g(y) by separating the x and y terms onto opposite sides and integrating each independently.
What is the difference between a general and particular solution?
A general solution contains arbitrary constants; a particular solution has specific values substituted using given initial conditions.