Concept Core
From classification to solution methods β the complete ODE framework for EAPCET.
Classification: Order & Degree
| Term | Definition | Example |
|---|
| Order | Highest derivative present | dΒ²y/dxΒ² + dy/dx = 0: order 2 |
| Degree | Power of the highest derivative (after clearing radicals/fractions) | (dy/dx)Β³ + y = 0: degree 3 |
Note: Degree is defined only when the ODE is a polynomial in derivatives. Equations with terms like e^(dy/dx) or sin(dy/dx) have undefined degree.
Variable Separable Method
When the ODE can be written as f(y)dy = g(x)dx, integrate both sides separately.
dy/dx = g(x)/h(y) β h(y)dy = g(x)dx β β«h(y)dy = β«g(x)dx + C
This is the simplest and most commonly tested method. Always check if separation is possible first.
Homogeneous Differential Equations
An ODE dy/dx = f(x,y) is homogeneous if f(tx, ty) = f(x,y) for all t. Equivalently, dy/dx = g(y/x) form.
Solution: Substitute y = vx β dy/dx = v + xΒ·dv/dx. This converts it to separable form in v and x.
Linear First-Order ODE
Form: dy/dx + P(x)y = Q(x)
Integrating factor (IF): ΞΌ = e^(β«P(x)dx)
Solution: y Γ IF = β«Q(x) Γ IF dx + C
Multiply both sides by IF. The LHS becomes d/dx[y Γ IF], which integrates directly.
Exact Differential Equations
M(x,y)dx + N(x,y)dy = 0 is exact if βM/βy = βN/βx.
Solution: F(x,y) = β«M dx + g(y), where g(y) is found using βF/βy = N.
Formation of Differential Equations
To form a DE from y = f(x, cβ, cβ, ...), differentiate and eliminate the arbitrary constants.
n arbitrary constants β differentiate n times β eliminate constants β nth order ODE.
EAPCET pattern: "Form the DE representing all circles of radius r" β differentiate the circle equation twice to eliminate a, b; keep r.
Formula Vault
Differential equations methods and solution forms.
Variable Separable
β«h(y)dy = β«g(x)dx + C
Separate and integrate both sides
Homogeneous Substitution
y = vx β dy/dx = v + xΒ·dv/dx
Reduces to separable in v, x
Linear ODE Integrating Factor
IF = e^(β«P dx)
For dy/dx + Py = Q
Linear ODE Solution
yΒ·(IF) = β«QΒ·(IF)dx + C
Multiply both sides by IF first
Exactness Condition
βM/βy = βN/βx
For M dx + N dy = 0
Order of DE
Highest derivative order
dΒ²y/dxΒ²: order 2
Degree of DE
Power of highest derivative
After clearing radicals/fractions
Bernoulli Equation
dy/dx + Py = QyβΏ
Substitute v = y^(1-n) to linearise
Worked Examples
5 problems β order/degree, separable, homogeneous, linear ODE, and a common trap.
EasyFind order and degree of: (dΒ²y/dxΒ²)Β³ + 5(dy/dx) = 0βΎ
Find the order and degree of the differential equation (dΒ²y/dxΒ²)Β³ + 5(dy/dx) = 0.
1
Highest derivative: dΒ²y/dxΒ² β Order = 2
2
Power of the highest derivative (dΒ²y/dxΒ²)Β³: Degree = 3
β Order = 2, Degree = 3
EasySolve: dy/dx = (1+yΒ²)/(1+xΒ²)βΎ
Solve the differential equation dy/dx = (1+yΒ²)/(1+xΒ²).
1
Separate: dy/(1+yΒ²) = dx/(1+xΒ²)
2
Integrate both sides: tanβ»ΒΉy = tanβ»ΒΉx + C
β tanβ»ΒΉy = tanβ»ΒΉx + C or tanβ»ΒΉy β tanβ»ΒΉx = C
MediumSolve the linear ODE: dy/dx + y/x = xΒ²βΎ
Solve dy/dx + (1/x)y = xΒ².
1
P(x) = 1/x β IF = e^(β«1/x dx) = e^(ln x) = x
2
Multiply: d/dx[xy] = xΒ³
3
Integrate: xy = xβ΄/4 + C β y = xΒ³/4 + C/x
β y = xΒ³/4 + C/x
EAPCET LevelSolve the homogeneous ODE: x dy/dx = y + x tan(y/x)βΎ
Solve x dy/dx = y + x tan(y/x).
1
Let y = vx β dy/dx = v + x dv/dx
2
LHS: x(v + x dv/dx) = vx + xΒ² dv/dx
4
Simplify: xΒ² dv/dx = x tan(v) β dv/tan(v) = dx/x β cot(v)dv = dx/x
5
Integrate: ln|sin v| = ln|x| + C β sin(y/x) = Ax (A = eαΆ)
β sin(y/x) = Ax
Trap QuestionDegree of ODE with e^(dy/dx) β students say degree = 1βΎ
Find the degree of: e^(dy/dx) + xΒ·dy/dx = y.
1
The trap: The term e^(dy/dx) is transcendental β it cannot be expressed as a polynomial in dy/dx.
2
Degree is defined only when the ODE is a polynomial in its derivatives.
3
Since e^(dy/dx) is NOT a polynomial in dy/dx, the degree is undefined (not defined).
4
Order = 1 (highest derivative is dy/dx), but degree cannot be determined.
β Order = 1, Degree = not defined (transcendental in derivatives)
Mistake DNA
3 differential equations errors from EAPCET distractor analysis.
π
Degree = Order (Confusing the Two)
Order = highest derivative present. Degree = power of that highest derivative. They are different concepts.
β Wrong
(dy/dx)Β³ + y = 0:
Order = 3, Degree = 3 β
(order = 1, degree = 3)
β Correct
Highest derivative: dy/dx β
Order = 1 β
Power of (dy/dx)Β³ = 3 β
Degree = 3 β
Order and degree are both 'about' the highest derivative, but measure different things. Order = which derivative (1st, 2nd, ...); Degree = what power is it raised to.
π§
Forgetting e^(β«P dx) for Integrating Factor
For linear ODE dy/dx + Py = Q, the integrating factor is e^(β«P dx), not just P or β«P.
β Wrong
dy/dx + (2/x)y = x:
IF = 2/x β
(that's just P, not IF!)
β Correct
IF = e^(β«2/x dx) = e^(2ln x)
= xΒ² β
Then multiply ODE by xΒ² β
IF = e raised to the INTEGRAL of P, not P itself. Compute β«P dx first, then exponentiate.
β
Adding Constant to Both Sides After Integration
When separating variables and integrating, only ONE constant of integration is needed (combine LHS and RHS constants into one C).
β Wrong
β«f(y)dy = β«g(x)dx:
F(y) + Cβ = G(x) + Cβ β
(two constants; redundant)
β Correct
F(y) = G(x) + C β
(C = Cβ β Cβ)
One arbitrary constant
for first-order ODE
Combining Cβ β Cβ = C is standard practice. A first-order ODE has exactly one arbitrary constant in its general solution.
Chapter Intelligence
DEs connect Calculus, Physics (Newton's laws, SHM) and Chemistry (rate equations).
EAPCET Weightage (2019β2024)
Variable separable method~7 Order and degree identification~5 High-Yield PYQ Patterns
Identify order and degreeSolve by variable separationLinear ODE: find IF and solveHomogeneous: substitute y=vxForm DE from given curve familyDegree undefined for transcendental DEs
Exam Strategy
- Order/degree questions: find the highest derivative (order), then look at its power (degree). Watch for transcendental terms β degree is undefined for e^(dy/dx), sin(dy/dx), etc.
- Variable separable: move all y and dy to one side, all x and dx to the other. Integrate both sides. Add only one constant C.
- Linear ODE: identify P(x) and Q(x) from dy/dx + Py = Q. Compute IF = e^(β«P dx). Multiply and integrate.
- Homogeneous: if dy/dx = f(y/x), substitute y = vx. The equation separates into v and x form.
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