The direct answer
How to Identify the Type of Differential Equation
| Type | Recognition Pattern | Method |
|---|---|---|
| Variable Separable | dy/dx = f(x)·g(y) | Separate and integrate |
| Homogeneous | dy/dx = f(y/x) | Substitute y = vx |
| Linear | dy/dx + Py = Q | Integrating factor e^(∫P dx) |
| Bernoulli | dy/dx + Py = Qyⁿ | Substitute v = y^(1-n) |
| Exact | ∂M/∂y = ∂N/∂x | Integrate M dx + N dy |
Variable Separable — The Foundation
Form: dy/dx = f(x)·g(y)
Steps:
1. Rearrange: dy/g(y) = f(x)dx
2. Integrate both sides: ∫dy/g(y) = ∫f(x)dx + C
Example: dy/dx = xy
dy/y = x dx → ln|y| = x²/2 + C → y = Ce^(x²/2)
🧠 Memory trick: "Get all y's with dy on one side, all x's with dx on the other, then integrate." This is the most basic and most tested method.
Homogeneous Equations — The y = vx Substitution
Recognition: dy/dx = f(y/x) — every term in numerator and denominator has the same total degree.
Steps:
1. Substitute y = vx, so dy/dx = v + x(dv/dx)
2. The equation becomes v + x(dv/dx) = f(v)
3. Rearrange: x(dv/dx) = f(v) − v
4. Separate variables: dv/(f(v) − v) = dx/x
5. Integrate and substitute v = y/x back
Example: dy/dx = (x² + y²)/(xy)
dy/dx = (x/y) + (y/x) = 1/(y/x) + (y/x). Let v = y/x → v + x(dv/dx) = 1/v + v → x(dv/dx) = 1/v → v dv = dx/x → v²/2 = ln|x| + C
🧠 Memory trick: "y = vx always works for homogeneous." Check if every term is of the same degree (x², y², xy are all degree 2).
Linear Differential Equations — Integrating Factor
Form: dy/dx + P(x)·y = Q(x)
Steps:
1. Identify P(x) and Q(x)
2. Calculate IF = e^(∫P dx)
3. Solution: y·(IF) = ∫Q·(IF) dx + C
Example: dy/dx + 2y = eˣ
P = 2, IF = e^(∫2dx) = e^(2x)
y·e^(2x) = ∫eˣ·e^(2x)dx = ∫e^(3x)dx = e^(3x)/3 + C
y = eˣ/3 + Ce^(-2x)
🧠 Memory trick: "Multiply by e^(∫P dx), then the left side becomes d/dx(y·IF)." This is the most powerful tool in differential equations.
Bernoulli's Equation — Reduce to Linear
Form: dy/dx + P·y = Q·yⁿ (n ≠ 0, 1)
Steps:
1. Divide by yⁿ: y⁻ⁿ(dy/dx) + P·y^(1-n) = Q
2. Substitute v = y^(1-n), so dv/dx = (1-n)y⁻ⁿ(dy/dx)
3. Equation becomes: (1/(1-n))·(dv/dx) + P·v = Q
4. This is now linear in v — solve with integrating factor
5. Substitute y^(1-n) = v back
🧠 Memory trick: "Divide by yⁿ, substitute v = y^(1-n), and it becomes linear." Bernoulli is the trickiest type — practice it until it's automatic.
Exact Differential Equations
Form: M(x,y)dx + N(x,y)dy = 0
Check: Exact if ∂M/∂y = ∂N/∂x
Solution: ∫M dx (treat y as constant) + ∫(terms of N not containing x) dy = C
If not exact, look for integrating factor:
IF = e^(∫((∂M/∂y − ∂N/∂x)/N)dx) if the expression depends only on x
IF = e^(∫((∂N/∂x − ∂M/∂y)/M)dy) if the expression depends only on y
Frequently Asked Questions
How do I identify the type of differential equation?
Identify the type by checking: (1) Can you separate x and y? → Variable separable. (2) Is it of the form dy/dx = f(y/x)? → Homogeneous. (3) Is it of the form dy/dx + Py = Q? → Linear. (4) Does it have yⁿ term? → Bernoulli. (5) Is ∂M/∂y = ∂N/∂x? → Exact.
What is the order and degree of a differential equation?
Order is the highest derivative present. Degree is the power of the highest derivative, after removing radicals and fractions. For example, (d²y/dx²)³ + dy/dx = 0 has order 2 and degree 3. If the equation has radicals, square both sides first to find the degree.
How do I solve a homogeneous differential equation?
For dy/dx = f(y/x), substitute y = vx, then dy/dx = v + x(dv/dx). This transforms it into a variable separable equation. Solve for v in terms of x, then substitute v = y/x back.
What is the integrating factor for a linear differential equation?
For dy/dx + Py = Q, the integrating factor (IF) is e^(∫P dx). The solution is y·(IF) = ∫Q·(IF) dx + C. This converts the left side into d/dx(y·IF).
What is Bernoulli's equation and how do I solve it?
Bernoulli's equation is dy/dx + Py = Qyⁿ. Divide by yⁿ, then substitute v = y^(1-n). This transforms it into a linear equation in v, solvable with an integrating factor.
What are the most common differential equation questions in JEE?
JEE frequently asks: (1) Solving variable separable equations, (2) Homogeneous equations with substitution y = vx, (3) Linear equations with integrating factor, (4) Finding order and degree, (5) Forming DE from a family of curves, (6) Application problems.
How do I form a differential equation from a family of curves?
Start with the general equation of the family. Differentiate to get dy/dx. Eliminate the constant using the original equation. For y = Cx²: dy/dx = 2Cx, C = y/x², so x(dy/dx) = 2y.
Explore more calculus resources
Master differential equations, then move to integration and limits.
Putting this into practice
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