Engineering Mathematics

Differential Equations

First-order linear ODEs (integrating factor) and second-order constant-coefficient equations via the auxiliary equation.

PART 1

Topic Breakdown & Traps

The Engineering Principle

A differential equation relates a function to its own rate of change. A first-order linear ODE

\frac{d y}{d x} + P (x) y = Q (x)

is solved by multiplying through by an integrating factor that turns the left side into an exact derivative. A second-order constant-coefficient ODE

a y^{''} + b y^{'} + cy = 0

is solved by guessing

y = e^{m x}

, which reduces it to an algebraic auxiliary (characteristic) equation

a m^{2} + bm + c = 0

; the nature of its roots (real distinct, repeated, or complex) dictates the form of the solution.

The Core Formula Matrix

First-order linear:

\frac{d y}{d x} + P (x) y = Q (x)

, integrating factor

μ = e^{\int P d x}

, solution

y μ = \int Q μ d x + C

.

Second-order homogeneous:

a y^{''} + b y^{'} + cy = 0

→ auxiliary equation

a m^{2} + bm + c = 0

.
• Real distinct roots

m_{1}, m_{2}

y = C_{1} e^{m_{1} x} + C_{2} e^{m_{2} x}

.
• Repeated root

m

y = (C_{1} + C_{2} x) e^{m x}

.
• Complex

α \pm i β

y = e^{α x} (C_{1} cos β x + C_{2} sin β x)

.

Separable:

\frac{d y}{d x} = g (x) \Rightarrow y = \int g (x) d x + C

, fix

C

from the initial condition.

The ‘IIT Traps’

⚠
**Integrating factor uses $\int P d x$ , not $\int Q d x$ .** $μ = e^{\int P d x}$ depends only on the coefficient of $y$ .
⚠
**Repeated roots need the extra $x$ .** A double root gives $(C_{1} + C_{2} x) e^{m x}$ ; writing $C_{1} e^{m x} + C_{2} e^{m x}$ collapses to one constant and is wrong.
⚠
Apply initial conditions after the general solution. Fix the constants only once the full general solution is assembled — not term by term.

PART 2

Progressive 3-Tier Question Suite

Q1BASIC1 Mark · MCQ

The integrating factor of the linear ODE

\frac{d y}{d x} + 2 y = x

is:

Q2MEDIUM2 Marks · NAT

For the ODE

y^{''} - 5 y^{'} + 6 y = 0

, the larger root of its auxiliary equation is ______. (Round off to two decimal places.)

Q3HARD2 Marks · NAT

Solve

y^{''} - 4 y = 0

with

y (0) = 1

and

y^{'} (0) = 0

. The value of

y (1)

is ______. (Round off to two decimal places.)