Engineering Mathematics I (IOE BE First Year): Syllabus Overview and Study Tips

Engineering Mathematics I (course code ENSH 101) is the first-semester mathematics course for IOE BE students. In the current IOE syllabus it has five units: derivatives and their applications, antiderivatives (integration) and their applications, ordinary differential equations, plane analytic geometry, and three-dimensional geometry. It has 3 lecture hours and 2 tutorial hours a week, and the theory marks are split into 40 for internal assessment and 60 for the final exam.

The same ENSH 101 course appears in Year I Part I of several BE programs, including Civil and Computer. The summary below is written in our own words from the official syllabus. For the exact wording, open the course syllabus linked from the IOE curriculum structure page. Older batches followed an earlier version of the syllabus, so check which version applies to you.

Engineering Mathematics I syllabus: units and hours

Unit	Main topics	Lecture hours
1. Derivatives and applications	Review of derivatives, mean value theorems, indeterminate forms and L'Hospital's rule, Taylor's and Maclaurin's series, asymptotes, pedal equations, curvature and radius of curvature	10
2. Antiderivatives and applications	Definite and indefinite integrals, differentiation under the integral sign, improper integrals, Beta and Gamma functions, area, arc length, volume and surface of revolution, centroid and moment of inertia	11
3. Ordinary differential equations	First-order equations (separable, homogeneous, linear, Bernoulli), higher degree and Clairaut's form, second-order linear equations with constant coefficients, Cauchy's equation, electric circuit and mass-spring models	10
4. Plane analytic geometry	Translation and rotation of axes, equation of a conic in Cartesian and polar form, identifying conics	4
5. Three-dimensional geometry	Straight line in symmetric and general form, coplanar lines, shortest distance, sphere and tangent planes, right circular cone and cylinder	10
Tutorials across all units		30

That adds up to 45 lecture hours and 30 tutorial hours. Units 1, 2 and 3 carry about two-thirds of the lecture time, so they deserve most of your practice.

What to study in each unit

Unit 1: derivatives and applications

Be fast with differentiation rules first; everything else depends on it. Then practise limits of 0/0 and ∞/∞ type with L'Hospital's rule, expanding functions like eˣ, sin x and log(1 + x) as Maclaurin series, finding asymptotes of curves, and the radius of curvature formula.

Unit 2: antiderivatives and applications

Revise standard integrals and substitution. Learn Beta and Gamma functions well, because many definite integrals with powers of sin and cos are solved quickly with them. Applications such as area, arc length and volume of revolution are formula-driven once you can sketch the curve.

Unit 3: ordinary differential equations

Learn to recognise the type of equation before solving: separable, homogeneous, linear, Bernoulli, Clairaut or second-order with constant coefficients. Each type has a fixed method. The circuit and spring-mass models are just these equations applied to physics.

Units 4 and 5: geometry

For conics, practise removing the xy term by rotating axes and then naming the curve. In three dimensions, the shortest distance between two skew lines and the equation of a sphere are common problems.

Unit 2 idea: the definite integral adds up thin strips of height y and width dx. For y = x² from 0 to 2 the area is 8/3.

Formula table for quick revision

Topic	Formula or rule
Power rule	d/dx (xⁿ) = n xⁿ⁻¹
L'Hospital's rule	If f/g gives 0/0 or ∞/∞, then lim f/g = lim f′/g′
Maclaurin series	f(x) = f(0) + x f′(0) + (x²/2) f″(0) + (x³/6) f‴(0) + ...
Radius of curvature	ρ = (1 + y₁²)^(3/2) / \|y₂\|, where y₁ = dy/dx and y₂ = d²y/dx²
Gamma function	Γ(n + 1) = n Γ(n); Γ(1) = 1; Γ(1/2) = √π
Beta function	B(m, n) = Γ(m) Γ(n) / Γ(m + n)
Arc length	s = ∫ √(1 + (dy/dx)²) dx
Volume of revolution about x-axis	V = π ∫ y² dx
Linear first-order ODE	dy/dx + Py = Q; integrating factor = e^(∫P dx)
Bernoulli's equation	dy/dx + Py = Qyⁿ; put z = y^(1 − n) to make it linear
Clairaut's equation	y = px + f(p), where p = dy/dx; general solution y = cx + f(c)
Identifying a conic	ax² + 2hxy + by² + 2gx + 2fy + c = 0 (non-degenerate): h² − ab < 0 ellipse, = 0 parabola, > 0 hyperbola
Sphere	x² + y² + z² + 2ux + 2vy + 2wz + d = 0; centre (−u, −v, −w), radius √(u² + v² + w² − d)

Worked example: a linear differential equation

Solve dy/dx + y/x = x².

Compare with dy/dx + Py = Q: here P = 1/x and Q = x².
Integrating factor = e^(∫(1/x) dx) = e^(ln x) = x.
Multiply through by x: x dy/dx + y = x³, and the left side is d(xy)/dx.
Integrate: xy = x⁴/4 + C.
So y = x³/4 + C/x.

Check: dy/dx = 3x²/4 − C/x², and y/x = x²/4 + C/x². Adding them gives x², which matches the right-hand side.

A study plan for the semester

This plan follows the lecture hours in the syllabus and assumes a semester of about 15 teaching weeks. Shift it to match your college's schedule.

Weeks	Unit	Target
1 to 3	Derivatives and applications	Solve 10 to 15 problems per topic; make your own formula sheet
4 to 6	Antiderivatives and applications	Master Beta and Gamma; sketch each curve before finding area or volume
7 to 9	Ordinary differential equations	Make a one-page chart of equation types and methods
10	Plane analytic geometry	Rotation of axes and naming conics
11 to 13	Three-dimensional geometry	Lines, shortest distance, sphere problems
14 to 15	Revision	Solve full past papers under time limits

Internal assessment carries 40 marks, so do not ignore assignments and class tests. They are easier marks than the final exam.

Useful links

Official course list and syllabus links: IOE bachelor curriculum page
Chapter-wise notes: BE notes hub
Past papers to practise: IOE old questions
Other first-year subjects: syllabus hub
After results: GPA and CGPA calculator

Frequently asked questions

How many units are in IOE Engineering Mathematics I?

The current syllabus has five units: derivatives and applications, antiderivatives and applications, ordinary differential equations, plane analytic geometry, and three-dimensional geometry.

How are marks divided in Engineering Mathematics I?

The current curriculum structure gives 40 marks for internal assessment and 60 for the final exam, a total of 100. There is no practical component.

Which unit is the hardest?

Students differ, but differential equations and Beta and Gamma functions take the most practice for many first-year students. Start them early rather than leaving them for the last weeks.