Differential Equations
1. Introduction
A differential equation is an equation involving derivatives of a function. Differential equations model physical phenomena including motion, heat transfer, population growth, and electrical circuits.
2. Basic Concepts
2.1 Order
The order of a differential equation is the highest derivative present. Example: d²y/dx² + 3 dy/dx + 2y = 0 is second order.
2.2 Degree
The degree is the highest power of the highest derivative, after removing radicals and fractions involving derivatives. Example: (d²y/dx²)² + (dy/dx)³ + y = 0 has degree 2.
'To find degree, first make sure the equation is a polynomial in derivatives. If it involves √(dy/dx), square both sides first.'
3. Formation of Differential Equations
Form by eliminating arbitrary constants from a given family of curves. The number of constants eliminated equals the order of the resulting DE.
4. Methods of Solution
4.1 Variable Separable
If dy/dx = f(x)g(y), separate to get (1/g(y)) dy = f(x) dx. Then integrate both sides.
4.2 Homogeneous Equations
If dy/dx = f(x, y)/g(x, y) where f and g are homogeneous of the same degree, substitute y = vx.
4.3 Linear Differential Equations
Form: dy/dx + P(x)y = Q(x). Integrating factor (IF) = e^{∫ P dx}. Solution: y · IF = ∫ Q · IF dx + C.
4.4 Exact Differential Equations
Form: M dx + N dy = 0, where ∂M/∂y = ∂N/∂x. Solution: ∫ M dx (treating y constant) + ∫ terms of N not containing x dy = C.
4. Applications of Differential Equations
4.1 Population Growth (Malthusian Model)
dP/dt = kP, where P is population and k is growth rate. Solution: P = P₀e^{kt}, where P₀ is initial population.
4.2 Radioactive Decay
dN/dt = -λN, where N is number of atoms and λ is decay constant. Solution: N = N₀e^{-λt}. Half-life: T_{1/2} = ln 2/λ.
4.3 Newton's Law of Cooling
dT/dt = -k(T - Tₘ), where Tₘ is the surrounding temperature. Solution: T - Tₘ = (T₀ - Tₘ)e^{-kt}.
4.4 Growth of Current in LR Circuit
L dI/dt + RI = E, where E is applied EMF. Solution: I = (E/R)(1 - e^{-Rt/L}).
5. Solving First Order DEs — A Systematic Approach
5.1 Step-by-Step Method
- Identify the type of equation (variable separable, homogeneous, linear, exact).
- For variable separable: Rearrange to f(y)dy = g(x)dx and integrate.
- For homogeneous: Write dy/dx = F(y/x), substitute y = vx.
- For linear: Find IF = e^{∫P dx}, multiply both sides, integrate.
- For exact: Check ∂M/∂y = ∂N/∂x, then integrate M dx + terms of N not containing x.
5.2 Common Forms and Their Substitutions
| Form | Substitution |
|---|---|
| dy/dx = f(ax + by + c) | u = ax + by + c |
| dy/dx = (ax + by + c)/(dx + ey + f) | Shift origin if lines intersect, or u = ax+by |
| Bernoulli: dy/dx + Py = Qyⁿ | v = y^{1-n} |
6. Worked Problems
Problem 1: Solve dy/dx = (x + y + 1)/(2x + 2y + 3). Solution: Let u = x + y. Then du/dx = 1 + dy/dx ⇒ dy/dx = du/dx - 1. So du/dx - 1 = (u + 1)/(2u + 3). Simplify and solve as variable separable.
Problem 2: Solve dy/dx + y cot x = 2 cos x. Solution: This is linear with P = cot x, Q = 2 cos x. IF = e^{∫ cot x dx} = e^{ln|sin x|} = sin x. y sin x = ∫ 2 cos x sin x dx = ∫ sin 2x dx = -cos 2x/2 + C. Hence y = (-cos 2x)/(2 sin x) + C/sin x.
Problem 3: Form the DE of the family of circles with center on the y-axis. Solution: x² + (y - k)² = r². Differentiating: 2x + 2(y - k)y' = 0. Differentiating again: 2 + 2(y - k)y'' + 2(y')² = 0. Eliminate (y - k). The DE is (1 + (y')²)y'' = xy''.
6. Common Mistakes
'Students often forget the constant of integration when solving DEs. The general solution must include an arbitrary constant.'
'When solving homogeneous equations, verify that the function is truly homogeneous before using the y = vx substitution.'
7. ISC Exam Focus
| Topic | Theory Marks | Practical Marks |
|---|---|---|
| Order and degree | 2 | 1 |
| Formation of DE | 2 | 1 |
| Variable separable | 3 | 2 |
| Homogeneous equations | 3 | 2 |
| Linear equations | 4 | 3 |
| Exact equations | 3 | 2 |
8. Self-Test Questions
- Find the order and degree of (d²y/dx²)² = (1 + (dy/dx)³)^{2/3}.
- Form the DE of all circles of radius 2.
- Solve: (x² - y²) dy/dx = 2xy.
- Solve: dy/dx + 2y tan x = sin x, y(0) = 0.
- Solve: (x² + y²) dx - 2xy dy = 0.
