Differential Equations
Learn Differential Equations with notes, examples, and practice questions.
Chapter Overview — Equations with Derivatives
An ordinary differential equation (ODE) has one independent variable (usually x). A partial differential equation has several — PDEs are beyond Class 12 scope.
A differential equation (DE) is an equation that connects an unknown function with one or more of its derivatives. The unknown is a function y = f(x), not a single number.
Solving a DE means finding all functions y(x) that satisfy the relation. Applications include population growth, cooling of objects, motion under resistance, and chemical reaction rates.
This chapter focuses on first-order, first-degree equations and three standard solution methods used in CBSE Class 12.
Order and Degree
Order = highest order derivative appearing in the equation (dy/dx, d²y/dx², …).
Degree = power of the highest-order derivative after clearing fractions and radicals — but only defined when the equation is polynomial in derivatives.
- dy/dx = 5x² → order 1, degree 1
- d²y/dx² + (dy/dx)³ = 0 → order 2, degree 1
- (dy/dx)² = 4y → order 1, degree 2
- (1 + (dy/dx)²)^(3/2) = d²y/dx² → degree not defined (fractional power)
General vs Particular Solution
- General solution — contains arbitrary constants (C, C₁, C₂, …) equal in number to the order of the DE
- Particular solution — obtained when initial/boundary conditions fix the constants
- Formation of DE — differentiate as many times as there are arbitrary constants, then eliminate constants
Method 1 — Variable Separable
Separable form
g(y) dy = f(x) dx → ∫g(y) dy = ∫f(x) dx + C
Example type
dy/dx = (1+x²)/(1+y²) → (1+y²) dy = (1+x²) dx
If the equation can be written as g(y) dy = f(x) dx (all y terms with dy, all x terms with dx), integrate both sides directly.
Sometimes a substitution or simple algebra (cross-multiplying) is needed first.
Method 2 — Homogeneous Equations
Substitution
y = vx, dy/dx = v + x·dv/dx
Test
Write F(x,y) = G(y/x) — if possible, equation is homogeneous
After substitution
v + x·dv/dx = G(v) → separate variables in v and x
A first-order DE dy/dx = F(x,y) is homogeneous if F(tx, ty) = F(x,y) for all t — equivalently, F can be written as a function of y/x alone.
Substitute y = vx, so dy/dx = v + x(dv/dx). This always converts a homogeneous equation into a separable equation in v and x.
Method 3 — Linear Differential Equations
Linear form
dy/dx + P(x)y = Q(x)
Integrating factor
IF = e^(∫P(x) dx)
Solution
y · IF = ∫ Q · IF dx + C
Special case
dy/dx + (1/x)y = Q(x) → IF = x
Standard linear form: dy/dx + P(x)·y = Q(x). Here P and Q are functions of x only.
Integrating factor (IF) = e^(∫P dx). Multiply the entire equation by IF — left side becomes d/dx [y · IF].
Solution: y · IF = ∫ Q · IF dx + C, then divide by IF.
How to Choose the Right Method
- Can you put all y with dy and all x with dx? → Variable separable
- Is dy/dx a function of y/x only (after rearranging)? → Homogeneous (y = vx)
- Is it in the form dy/dx + Py = Q? → Linear (integrating factor)
- Check degree 1 in dy/dx — all three methods need first-order equations
Formation of Differential Equations
Given a family of curves with n arbitrary constants, differentiate n times and eliminate all n constants. The result is an nth-order differential equation whose general solution is that family.
Example: y = C e^x → dy/dx = C e^x = y → DE is dy/dx = y.
Solved Examples
Step-by-step solutions — read each step before checking the final answer.
Example 1: Variable Separable
Solve dy/dx = xy.
- 1Separate: dy/y = x dx.
- 2Integrate both sides: ln|y| = x²/2 + C₁.
- 3Exponentiate: y = C e^(x²/2), where C = ±e^C₁.
Answer
y = C e^(x²/2)
Example 2: Separable with Initial Condition
Solve dy/dx = 2y, given y(0) = 3.
- 1dy/y = 2 dx → ln|y| = 2x + C.
- 2y = C e^(2x). At x = 0: 3 = C → C = 3.
Answer
y = 3e^(2x)
Example 3: Homogeneous Equation
Solve dy/dx = (x + y)/x.
- 1RHS = 1 + y/x → homogeneous. Put y = vx, dy/dx = v + x·dv/dx.
- 2v + x·dv/dx = 1 + v → x·dv/dx = 1 → dv = dx/x.
- 3v = ln|x| + C → y/x = ln|x| + C.
Answer
y = x(ln|x| + C)
Example 4: Linear DE
Solve dy/dx + 2y = e^x.
- 1P(x) = 2, Q(x) = e^x. IF = e^(∫2 dx) = e^(2x).
- 2Multiply: e^(2x)·dy/dx + 2e^(2x)·y = e^(3x).
- 3Left side = d/dx[y·e^(2x)]. So y·e^(2x) = ∫e^(3x) dx = e^(3x)/3 + C.
- 4y = e^x/3 + C·e^(−2x).
Answer
y = (1/3)e^x + C e^(−2x)
Example 5: Linear with 1/x
Solve x·dy/dx + y = x².
- 1Divide by x: dy/dx + (1/x)y = x.
- 2P = 1/x, Q = x. IF = e^(∫(1/x)dx) = e^(ln x) = x.
- 3x·dy/dx + y = x² → d/dx(xy) = x² → xy = x³/3 + C.
Answer
y = x²/3 + C/x
Example 6: Form a DE
Form the differential equation for y = C₁e^x + C₂e^(−x).
- 1y′ = C₁e^x − C₂e^(−x).
- 2y″ = C₁e^x + C₂e^(−x) = y.
- 3Eliminating constants: d²y/dx² = y.
Answer
d²y/dx² − y = 0 (order 2)
Key Points to Remember
- ✓ Order = highest derivative; degree = power of highest derivative (when defined)
- ✓ General solution has n constants for order n
- ✓ Variable separable: collect dy with y and dx with x
- ✓ Homogeneous: substitute y = vx to reduce to separable form
- ✓ Linear: IF = e^(∫P dx), then y·IF = ∫Q·IF dx + C
- ✓ Always verify by substituting your answer back into the DE
Exam Tips
- • State the method name before solving (e.g. 'This is a linear DE')
- • For linear DEs, write P, Q, and IF explicitly — step marks matter
- • Don't forget + C in general solution; use given point to find C for particular solution
- • Homogeneous check: can you write RHS as f(y/x)?
- • Formation problems: differentiate and eliminate — count constants carefully
- • Board often combines formation + solving in two-part questions
Formula Cheat Sheet
Quick reference — NCERT board exam formulas
Differential Equations
- Separable:dy/dx = f(x)g(y) ⇒ ∫dy/g(y) = ∫f(x)dx + C
- Homogeneous:Substitute y = vx
- Linear:dy/dx + P(x)y = Q(x)
- IF = e^(∫P dx)
- Solution: y·IF = ∫Q·IF dx + C
Practice MCQs — Differential Equations
Test your understanding with topic-wise multiple choice questions. Explanations appear after each answer.
The order of d²y/dx² + (dy/dx)³ = sin x is: