Higher Order Differential Equations

🟢 Lite — Quick Review (1h–1d)

Core Methods

Type	Method	Key Step
Linear with constant coeff.	Characteristic equation (CE)	Replace y → r, d²y/dx² → r²
Homogeneous eqn	Solve CE: ar² + br + c = 0	Real roots → exponentials; repeated → multiply by x
Non-homogeneous	CF + PI	Guess PI based on RHS form
Cauchy-Euler	Assume x^m	Replace x^r pattern
Variation of parameters	CF → u₁, u₂ → Wronskian	PI = −y₁∫y₂R/W + y₂∫y₁R/W

⚡ Quick记住: Real distinct roots → e^(r₁x), e^(r₂x); Real equal r → e^(rx), xe^(rx); Complex r = α ± iβ → e^(αx)[cos βx, sin βx]

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🟡 Standard — Regular Study (2d–2mo)

Linear DEs with Constant Coefficients

Homogeneous Case

Form: a d²y/dx² + b dy/dx + cy = 0

Characteristic Equation (CE): ar² + br + c = 0

CE Roots	CF (Complementary Function)
r₁ ≠ r₂ real	C₁e^(r₁x) + C₂e^(r₂x)
r₁ = r₂ = r	C₁e^(rx) + C₂xe^(rx)
r = α ± iβ	e^(αx)[C₁cos βx + C₂sin βx]

For n-th order: CE is degree n polynomial. Sum over all root contributions.

Non-Homogeneous Case

Form: a y” + b y’ + c y = R(x)

General Solution: y = CF + PI

Particular Integral (PI) — Method of Undetermined Coefficients:

R(x) form	Trial PI form
e^(kx)	Ae^(kx) (if k not root of CE); else multiply by x^s where s = multiplicity
polynomial in x	General polynomial of same degree
sin(kx) or cos(kx)	A cos(kx) + B sin(kx)

⚡ GATE Common Mistake: If the forcing term R(x) matches a term in CF, multiply the trial PI by x (or x² if needed). This is the “collision” rule.

Method of Variation of Parameters

For y” + P(x)y’ + Q(x)y = R(x):

1. Find two linearly independent solutions of homogeneous: y₁, y₂
2. Wronskian: W = y₁y₂’ − y₁’y₂
3. PI: y_p = −y₁∫(y₂R/W) dx + y₂∫(y₁R/W) dx

Advantage over undetermined coefficients: Works for ANY R(x) form.

Cauchy-Euler Equation

Standard form: x²y” + axy’ + by = R(x)

Trial solution: Assume y = x^m → get auxiliary equation: m(m−1) + am + b = 0

Roots of m	CF
m₁, m₂ real distinct	C₁x^(m₁) + C₂x^(m₂)
m₁ = m₂ = m	C₁x^m + C₂x^m ln x
complex m = α ± iβ	x^α[C₁cos(β ln x) + C₂sin(β ln x)]

⚡ Key difference from constant coefficient: Powers of x replace exponentials. When x > 0, ln x is defined.

Non-homogeneous Cauchy-Euler: Use variation of parameters or guess method adapted for x^m forms.

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🔴 Extended — Deep Study

Detailed Solution Process — Constant Coefficients

Example (GATE 2021): y” − 3y’ + 2y = e^(2x)

1. Homogeneous: r² − 3r + 2 = 0 → (r−1)(r−2) = 0 → r = 1, 2 → CF = C₁e^x + C₂e^(2x)

2. PI: R(x) = e^(2x), and r=2 IS a root of CE (collision with CF) → Trial: y_p = Ax·e^(2x) → substitute and solve → After substitution: A = 1 (working through derivatives) → y_p = x·e^(2x)

3. General: y = C₁e^x + C₂e^(2x) + x·e^(2x)

⚡ Collision Detection: e^(2x) appears in CF with C₂ term. Since it collides, multiply trial by x. If r=2 had multiplicity 2, multiply by x².

Wronskian and Linear Independence

Wronskian W(y₁,y₂): $$W = \begin{vmatrix} y_1 & y_2 \ y_1’ & y_2’ \end{vmatrix} = y_1 y_2’ - y_1’ y_2$$

• If W ≠ 0 at some point → solutions are linearly independent
• Used in variation of parameters formula

Example: y₁ = e^x, y₂ = xe^x → W = e^x·e^x + e^x·xe^x = e^(2x)(1 + x) ≠ 0 → LI ✓

Variation of Parameters — Worked Example

Problem: y” + y = tan x

1. Homogeneous: r² + 1 = 0 → r = ±i → y₁ = cos x, y₂ = sin x
2. W = y₁y₂’ − y₁’y₂ = cos²x + sin²x = 1
3. PI = −cos x∫(sin x·tan x/1) dx + sin x∫(cos x·tan x/1) dx = −cos x∫(sin²x/cos x) dx + sin x∫(sin x) dx = −cos x∫(sin²x/cos x) dx − sin x cos x = −cos x[−sin x + ln|sec x + tan x|] − sin x cos x = cos x·sin x − cos x·ln|sec x + tan x| − sin x cos x = −cos x·ln|sec x + tan x|

Second-Order Reduction of Order

If one solution y₁ is known for y” + Py’ + Qy = 0, the second solution is: $$y_2 = y_1 \int \frac{e^{-\int P dx}}{y_1^2} dx$$

GATE use: Sometimes gives y₁ = e^x and asks to find general solution — this formula gets the second linearly independent solution.

GATE Previous-Year Highlights

Year	Problem	Key Concept
2018	y” − 6y’ + 9y = e^(3x)	Repeated root r=3 (multiply by x²)
2019	Cauchy-Euler: x²y” − 2xy’ + 2y = x³	m² − 3m + 2 = 0 → m=1,2
2020	y” + y = sec x	Variation of parameters needed
2021	y” + 4y = sin 2x	Collision: PI needs x·(A cos 2x + B sin 2x)
2022	y” + y’ + y = 0, find Wronskian	W = Ce^(−x) — always nonzero
2023	Cauchy-Euler: x²y” + 5xy’ + 4y = 0	m² + 4m + 4 = 0 → m = −2 (repeated)

⚡ GATE Trap: In Cauchy-Euler, always check if m is repeated — that ln x term is frequently forgotten, costing 2 marks.

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