What is a differential equation? Order and degree
A differential equation contains an unknown function y and one or more of its derivatives, like dy/dx = 2x or y″ + y = 0.
Order = the highest derivative that appears. Degree = the power of that highest derivative, after the equation is made free of radicals and fractions in the derivatives. The degree is defined only if the equation is a polynomial in the derivatives.
- dy/dx + y = 0: order 1, degree 1.
- (d²y/dx²)³ + (dy/dx)⁴ = x: order 2, degree 3.
- y‴ + sin(y′) = 0: order 3, degree not defined.
General and particular solutions
A solution is a function that makes the equation true. The general solution has as many arbitrary constants as the order, e.g. y = Ceˣ for dy/dx = y. A particular solution has no arbitrary constants; we find them from given conditions like y(0) = 1, giving y = eˣ.
To check a solution, substitute it (and its derivatives) into the equation.
Variables separable
If dy/dx = g(x)·h(y), write dy/h(y) = g(x) dx and integrate both sides:
∫dy/h(y) = ∫g(x) dx + C.
Example: dy/dx = −x/y → y dy = −x dx → y²/2 = −x²/2 + c → x² + y² = C (a family of circles).
Homogeneous differential equations
F(x, y) is homogeneous of degree 0 if F(λx, λy) = F(x, y); then dy/dx = F(x, y) can be written as a function of y/x.
- Put y = vx, so dy/dx = v + x dv/dx.
- Substitute and simplify; x and v now separate.
- Integrate, then put back v = y/x.
If the equation looks like dx/dy = G(x/y), use x = vy instead.
Linear first-order differential equations
Form: dy/dx + P(x) y = Q(x).
- Integrating factor: I.F. = e^∫P dx.
- Solution: y × I.F. = ∫(Q × I.F.) dx + C.
Example: dy/dx + y = x. I.F. = eˣ. y eˣ = ∫x eˣ dx = eˣ(x − 1) + C → y = x − 1 + Ce⁻ˣ.
For dx/dy + P₁(y) x = Q₁(y), swap roles: I.F. = e^∫P₁ dy and x × I.F. = ∫(Q₁ × I.F.) dy + C.
Key formulas and definitions
- Order = highest derivative; degree = its power (polynomial form)
- Separable: ∫dy/h(y) = ∫g(x) dx + C
- Homogeneous: y = vx, dy/dx = v + x dv/dx
- Linear: dy/dx + Py = Q, I.F. = e^∫P dx
- y · I.F. = ∫Q · I.F. dx + C
Worked examples
1. Find the order and degree of (d²y/dx²)² + 3(dy/dx)³ + y = 0.
Highest derivative d²y/dx², so order 2. Its power is 2, so degree 2.
2. Show that y = Ae²ˣ is a solution of dy/dx = 2y.
dy/dx = 2Ae²ˣ = 2y. It satisfies the equation, so it is a solution (general, since it has one constant).
3. Solve dy/dx = (1 + y²)/(1 + x²).
dy/(1 + y²) = dx/(1 + x²). tan⁻¹y = tan⁻¹x + C.
4. Solve dy/dx = e^(x − y), given y(0) = 0.
eʸ dy = eˣ dx → eʸ = eˣ + C. y(0) = 0: 1 = 1 + C → C = 0. So eʸ = eˣ, i.e. y = x.
5. Solve the homogeneous equation dy/dx = (x + y)/x.
y = vx: v + x dv/dx = 1 + v → x dv/dx = 1 → v = ln|x| + C → y = x ln|x| + Cx.
6. Solve the homogeneous equation dy/dx = (x² + y²)/(2xy).
y = vx: v + x dv/dx = (1 + v²)/(2v) → x dv/dx = (1 − v²)/(2v) → 2v dv/(1 − v²) = dx/x → −ln|1 − v²| = ln|x| + c → x(1 − v²) = C → x² − y² = Cx.
7. Solve dy/dx + 2y = e⁻ˣ.
P = 2, I.F. = e²ˣ. y e²ˣ = ∫e⁻ˣ e²ˣ dx = eˣ + C. y = e⁻ˣ + Ce⁻²ˣ.
8. Solve dy/dx + y/x = x² (x > 0), given y(1) = 1.
I.F. = e^∫(1/x)dx = x. xy = ∫x³ dx = x⁴/4 + C. y(1) = 1: 1 = 1/4 + C → C = 3/4. y = x³/4 + 3/(4x).
Common mistakes
- Giving a degree to an equation like sin(y′) = y; the degree is not defined because it is not a polynomial in y′.
- Forgetting + C, or adding a separate constant on both sides (one C is enough).
- In linear equations, not first writing the equation as dy/dx + Py = Q (dividing by the coefficient of dy/dx).
- In homogeneous equations, forgetting to put back v = y/x at the end.