Engineering Mathematics

Ordinary Differential Equations

First-order, higher-order linear and Euler–Cauchy equations — the differential equations that model beams, flow and decay.

PART 1

Topic Breakdown & Traps

An ODE relates a function to its derivatives. A first-order linear equation

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

is solved with the integrating factor

e^{\int P d x}

. A homogeneous linear equation with constant coefficients is solved from its characteristic (auxiliary) equation: each real root

m

contributes

e^{m x}

, repeated roots add an

x

factor, and complex roots give

e^{α x} (cos β x, sin β x)

Variable separable:

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

Integrating factor:

μ = e^{\int P d x}

for

y^{'} + P y = Q

Auxiliary equation:

a y^{''} + b y^{'} + cy = 0 \Rightarrow a m^{2} + bm + c = 0

Distinct real roots:

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

⚠
Apply the initial condition last. Solve the general solution first, then fix the constants — never before integrating.
⚠
Repeated roots. A double root $m$ gives $(C_{1} + C_{2} x) e^{m x}$ , not $C_{1} e^{m x} + C_{2} e^{m x}$ .
⚠
Order ≠ degree. Order is the highest derivative; degree is its power after the equation is made polynomial in derivatives.

Advanced Engineering Mathematics — Erwin Kreyszig · Ordinary Differential Equations
Higher Engineering Mathematics — B.S. Grewal

PART 2

Q1BASIC1 Mark · MCQ

The general solution of

\frac{d y}{d x} = k y

is:

Q2MEDIUM2 Marks · NAT

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

with

y (0) = 1

, then

y (2)

is _____.

Q3HARD2 Marks · MCQ

The general solution of

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

is: