linear algebra vs differential equations:(explained)

In this post, you will discover the difference between linear algebra and differential equations, in an easy and appealing way. So keep reading to not miss the steps.

The difference between linear algebra and differential equations

Differential equations is a subject that is usually taken by engineering fields like mechanical and aerospace engineers.  while linear algebra is used frequently in computer science positions such as google ranking algorithm and other programming applications.

Differential equations are more difficult than linear algebra because it contains a lot of calculus applications such as derivatives and integrals.  In addition trigonometry functions like sinus cosine etc.

Linear algebra is a subject that’s Based on mathematic algebra foundations like arithmetics vectors and spaces. But the most interesting subject that you would be focusing on studying linear algebra is matrices, you would be using a lot of matrices. 

In this post are going to discover the full curriculum of both differential equations and linear algebra. We’re not going to dig deeper into the complex subjects but just give a simple explanation to keep you close to the information no what it takes to study these two subjects.

what do students study in differential equations

the differential equation is an equation that has at least one derivative variable or more. differential equations are used hugely in engineering occupations.

Normally the basic equation differential is written in the form below.

That is to say, you will be solving the kind of equations that we mentioned in the image above. To solve these equations you have to be good at derivatives because to solve any differential equation you will need to do a lot of derivatives

the second important thing in the differential that you should be good at is integrals. In differential equations, you will be studying and using a lot of integral functions to solve differential equations.

Normally differential equations are divided into 3 principals parts:

  • First-order differential equations
  • Second-order differential equations
  • Laplace transform and systems

1 – First-order Differential Equations

The first-order differential equation is written in this form below:

normally the solution of this form of a differential equation is in this form below.

Y=1/I(X) [ ∫ I(x)P(x)dx +c]

y’ + P(x)y=Q(x) and I(x)=exp( P(x))dx

it might be confusing a little bit, but the principle is to show or give a close approximation of what you will be studying in this first part in first-order differential equations.

if you want to know how to solve this equation you could watch this full procedure by watching this video.

In first-order equations you will be studying all theses following subjects:

  • 1) Verifying solutions
  • 2) four fundamental equations
  • 3) Classifying differential equations
  • 4) Basic Integration
  • 5) Separation of variable method
  • 6) Integration factor method
  • 7) Direct substitution method
  • 8) Homogeneous equation
  • 9) Bernoulli’s equation
  • 10) Exact equation
  • 11) Almost-exact equation
  • 12) Numerical Methods
  • 13) Euler’s method
  • 14) Runge-Kutta method
  • 15) Directional fields
  • 16) Existence & Uniqueness Thm
  • 17) Autonomous equation

So in general in the first-order differential equation, you have to be good in derivatives and integrals. That being said good and have succeeded in calculus1 and calculus 2 courses to know find issues in this subject.

Second-Order Differential Equations

The Second-order differential equation is written in this form below:

we call it a second-order differential equation because it has a double derivative, not like the first degree equations. Normally second-order differential equations are much more complex to solve than the first one.

to solve this equation you will need to execute multiple operations. In this case, you will be also using some trigonometry functions like cousins and sinus to find a solution to these equations.

if you are interested and want to know how to solve this equation you could watch this detailed video.

In this step, you will go further and study some advanced subjects that we’re going to list below to solve the second differential equation:

  • 1) 2nd Order Linear Differential Eq.
  • 2) Reduction of Order Method
  • 3) Constant Coefficient Diff. Eq
  • 4) Cauchy-Euler Diff. Equation
  • 5) Higher-Order Constant Coefficient Eq
  • 6) Non-homogeneous Diff. Eq
  • 7) Undetermined Coefficient Method
  • 8) Variation of Parameters Method

To not find issues with second-order differential equations you will need to have solid basics in trigonometry functions and exponents. There are used hugely in this case.

Series, Laplace Transforms & Systems

the last and the most complicated subject in differential equations is the Laplace transform. the best thing to explain Laplace’s transformation is by watching the video below:

So in Laplace transform, you will be studying all these subjects that we’re going to list below.

  • 1) Series Solution Method
  • 2) SeeSaw method for series
  • 3) Laplace transform method
  • 4) Find Laplace transform
  • 5) Inverse Laplace transform 7:37:9
  • 6) Cover-Up Method
  • 7) Solving Diff. Equations
  • 8) Convolution method
  • 9) Heaviside function
  • 10) Dirac Delta function
  • 10) System of equations
  • 11) Elimination method
  • 12) Laplace transform method
  • 13) Eigenvectors method
  • 14) Shortcut method for 2 x 2
  • 15) Shortcut method for 3 x 3 (repeat)
  • 16) Shortcut method for 3 x 3

In Laplace transformation you must be good in the all the following math subjects:

  • derivatives
  • integrals
  • trigonometry functions
  • complex numbers

that being said all the subjects tath you have students in calculus 2.

if you are interested to study a full curriculum on your own you could watch this free full course of differential equations on youtube.

what do students study in linear algebra?

in linear algebra, you will be focused on two interesting part vectors and matrices. That being said in linear algebra you will be studying a lot of functions like in this image below.

In addition, you will be solving linear systems like in the example below

3x+4y+8z=8

4y+3=-1

z=1

in linear algebra you will be studying 2 principal parts of chapters that are:

  • vector in space
  • Isomorphism

to explain linear algebra we will use video to make the idea clear because it is very hard to explain as written words. videos have animations are better in this case.

part 1 Vectors in Space

in this chapter you will be studying theses following subejcts:

  1. Describing Solution Sets, Part One
  2. Vector Length and Angle Measure
  3. Gauss-Jordan Elimination
  4. The Linear Combination Lemm
  5. Vector Spaces
  6. Subspaces
  7. Linear Independence
  8. Basis
  9. Dimension
  10. Vector Spaces and Linear Systems

part 2: Isomorphism

  1. Isomorphism
  2. Dimension Characterizes Isomorphism
  3. Homomorphism
  4. Range Space and Null Space
  5. Extra Transformations of the Plane
  6. Representing Linear Maps
  7. Matrix Represents a Linear Map
  8. Sums and Scalar Products of Matrices
  9. Matrix Multiplication, Part One

To study linear algebra you have to be good at the basics in algebra 1 and algebra 2 courses, especially vectors and matrices. But in general linear algebra is not complicated as differential equations.

if you are interested to discover a full course of linear algebra you could watch this free full learn algebra course.

Conclusion

it is recommended to learn linear algebra first and then go to the next step which is differential equations because you will need a lot of linear algebra subjects to use in differential equations. especially theses following subjects:

  • Vector spaces
  • Bases
  • Linear mappings
  • Matrices
  • Eigenvalues and eigenvectors
  • The characteristic and minimal polynomial of a matrix
  • Jordan canonical form