# IIT Delhi MTL1002 Linear Algebra and Differential Equations Guide

> Study guide for IIT Delhi MTL1002 Mathematics II, 2025 curriculum: linear algebra, first and higher order ODEs, linear systems and Laplace transforms.

Source: https://practicecalculus.com/courses/iit-delhi-mtl1002
Updated: 2026-11-08

## At a glance

- **Course code:** MTL1002 Mathematics II Linear Algebra and Differential Equations
- **Curriculum:** IIT Delhi curriculum, 2025 onwards
- **Credits:** 4 credits (L-T-P 3-1-0)
- **Offered by:** Department of Mathematics
- **Prerequisites:** None listed in the course template
- **Covers:** Linear systems, vector spaces, linear transformations, eigenvalues, first order ODEs and Picard's theorem, higher order linear ODEs, systems of ODEs, Taylor series method and Laplace transforms
- **Previous course:** [MTL1001 Mathematics I Calculus](/courses/iit-delhi-mtl1001)
- **Practice here:** [Differential equations](/learn/calculus-2/improper-integrals-and-applications/separable-equations) in Calculus 2

## Key takeaways

- MTL1002 Mathematics II is the second first year mathematics course in IIT Delhi's curriculum for 2025 onwards.
- It carries 4 credits with 3 lectures and 1 tutorial a week.
- Roughly the first half is linear algebra and the second half is ordinary differential equations.
- The differential equations part uses calculus throughout: integrating factors, Taylor series and the Laplace transform, which is an improper integral.

**MTL1002 Mathematics II Linear Algebra and Differential Equations** is the second first year mathematics course at the **Indian Institute of Technology Delhi (IIT Delhi)** in the curriculum for **2025 onwards**. It carries 4 credits, with 3 lectures and 1 tutorial a week, and pairs with [MTL1001 Mathematics I Calculus](/courses/iit-delhi-mtl1001). This guide follows the official course template on the IIT Delhi curriculum site and focuses on the calculus you need for the differential equations half.

## What MTL1002 covers

**Linear algebra.** Systems of linear equations, matrices and Gauss elimination, elementary row operations, row reduced echelon form, row space, column space, null space and rank; vector spaces over $\mathbb{Q}$, $\mathbb{R}$ and $\mathbb{C}$, subspaces, independence, span, basis, dimension, sums and direct sums; inner products and norms; linear transformations, the rank-nullity theorem, matrix representation, change of basis and similarity; eigenvalues, eigenvectors, the characteristic polynomial and the Cayley-Hamilton theorem (without proof).

**Differential equations.** A review of first order equations, existence and uniqueness for initial value problems and Picard's theorem; linear independence of solutions, the dimension of the solution space and the Wronskian; linear ODEs with constant coefficients of second and higher order, Cauchy-Euler equations, the method of undetermined coefficients and variation of parameters; systems of linear ODEs with constant coefficients, the fundamental matrix and matrix methods; the Taylor series method; and the Laplace transform and its use in solving ODEs.

The course outcomes include using row reduction to find inverses, ranks and solutions of linear systems, diagonalising linear transformations, and modelling physical phenomena with differential equations solved using linear algebra.

## MTL1002 topic map (calculus side)

PracticeCalculus does not teach linear algebra, so the table covers the differential equations topics and the calculus they rely on.

| MTL1002 topic | Practice | Read |
| --- | --- | --- |
| First order ODEs | [Separable equations](/learn/calculus-2/improper-integrals-and-applications/separable-equations) | [Differential equations practice](/practice-problems/differential-equations) |
| Direction of solutions and initial value problems | [Slope fields](/learn/calculus-2/improper-integrals-and-applications/slope-fields) | [Implicit differentiation](/blog/implicit-differentiation) |
| Growth and decay models | [Growth and decay](/learn/calculus-2/improper-integrals-and-applications/growth-and-decay) | [Exponential functions](/functions/exponential-functions) |
| Logistic and nonlinear models | [Logistic growth](/learn/calculus-2/improper-integrals-and-applications/logistic-growth) | [Natural exponential function](/functions/natural-exponential-function) |
| Taylor series method | [Taylor polynomials](/learn/calculus-2/power-series-parametric-polar/taylor-polynomials) | [Taylor series](/blog/taylor-series) |
| Numerical stepping (compare with Taylor method) | [Euler's method](/learn/calculus-2/improper-integrals-and-applications/eulers-method) | [Linear approximation](/blog/linear-approximation) |
| Laplace transform as an improper integral | [Infinite limits of integration](/learn/calculus-2/improper-integrals-and-applications/infinite-limits-of-integration) | [Improper integrals](/blog/improper-integrals) |
| Integration by parts for transforms | [Integration by parts](/learn/calculus-2/integration-techniques/integration-by-parts) | [Integral of x e^x](/integrals/x-e-to-the-x) |
| Partial fractions for inverse transforms | [Partial fractions](/learn/calculus-2/integration-techniques/partial-fractions) | [Partial fractions](/blog/partial-fractions) |
| Trig and exponential forcing terms | [Trig and exponential integrals](/learn/calculus-1/integrals/trig-exponential-integrals) | [Integral of e^x sin x](/integrals/e-to-the-x-sin-x) |

## Example problems at the MTL1002 level

These are original practice problems.

**1. A constant coefficient ODE.** For $y'' - 3y' + 2y = 0$, the characteristic equation $r^2 - 3r + 2 = 0$ has roots $1$ and $2$, so $y = C_1 e^{x} + C_2 e^{2x}$.

**2. A Laplace transform.** For $s > a$,

$$\mathcal{L}\{e^{at}\} = \int_0^\infty e^{-st}e^{at}\,dt = \lim_{T \to \infty}\left[\frac{e^{-(s - a)t}}{-(s - a)}\right]_0^T = \frac{1}{s - a}$$

**3. A Wronskian.** For $y_1 = \cos x$ and $y_2 = \sin x$, $W = y_1 y_2' - y_1' y_2 = \cos^2 x + \sin^2 x = 1 \ne 0$, so the solutions are independent.

## How to do well in MTL1002

- **Keep your integration fluent.** Laplace transforms are improper integrals, and inverse transforms usually need partial fractions.
- **Link the two halves.** The solution space of a linear ODE is a vector space, and the Wronskian is a test for linear independence, so the linear algebra makes the ODE theory make sense.
- **Use the weekly tutorial** to practise full solutions, including checking initial conditions at the end.
- **Practise in short daily sessions** on [practice](/practice) and revise from the [formula sheet](/formulas).

## Frequently asked questions

### What is MTL1002 at IIT Delhi?

MTL1002 Mathematics II Linear Algebra and Differential Equations is a 4 credit first year course (3 lectures and 1 tutorial a week) offered by the IIT Delhi Department of Mathematics in the curriculum for 2025 onwards.

### Which differential equations topics are in MTL1002?

The template lists a review of first order ODEs, existence and uniqueness and Picard's theorem, linear independence of solutions and the Wronskian, constant coefficient equations of second and higher order, Cauchy-Euler equations, undetermined coefficients, variation of parameters, systems with constant coefficients and the fundamental matrix, the Taylor series method, and the Laplace transform.

### Is MTL1002 a calculus course?

Not mainly. It is linear algebra and differential equations, but the ODE half uses calculus constantly, from integrating factors to improper integrals in the Laplace transform. PracticeCalculus covers that calculus side, not the linear algebra.

### How do I prepare for the ODE part of MTL1002?

Make sure integration techniques are fluent, especially integration by parts and partial fractions, because they appear in nearly every ODE and Laplace transform problem. Then practise separable and linear first order equations.