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Learn Differential Equations

Model how things change, from cooling coffee to vibrating springs.

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A differential equation relates a function to its own rates of change, which makes it the natural language for modeling anything that evolves: populations, cooling objects, circuits, springs, chemical reactions and orbits. An ordinary differential equations course covers first-order equations (separable, linear and exact), slope fields and Euler's method, second-order linear equations with constant coefficients, forcing and resonance, Laplace transforms, and systems of equations solved with eigenvalues. Learnr builds a personalized differential equations course and teaches it in short interactive lessons that connect each solution method to the physical behavior it describes.

How to learn Differential Equations, step by step

Review integration and derivatives. Practice u-substitution, integration by parts, partial fractions and derivatives of exponential and trig functions. Solving differential equations constantly requires integrating, so weak integration slows everything.
Learn first-order methods and slope fields. Solve separable equations, then linear equations with an integrating factor. Sketch slope fields to see the shape of solutions before solving, and use Euler's method to approximate one numerically.
Model real situations. Set up equations for exponential growth, logistic population growth, Newton's law of cooling and tank mixing problems. Writing the equation from words is the core skill, and checking units catches many setup errors.
Master second-order linear equations. Use the characteristic equation for constant coefficient equations and handle real, repeated and complex roots. Then add forcing terms with undetermined coefficients or variation of parameters, and study damping and resonance in a spring-mass system.
Learn Laplace transforms. Use a table of transforms to turn an initial value problem into algebra, solve, then invert with partial fractions. Laplace methods handle step functions and impulses that are awkward otherwise. Brief Learnr reviews help keep the transform pairs memorable.
Solve systems with linear algebra. Write coupled equations as x' = Ax, find eigenvalues and eigenvectors, and sketch phase portraits classified as nodes, saddles, spirals and centers. This links differential equations to the linear algebra you know.

What you'll learn

Classify differential equations by order, linearity and type
Solve separable, linear and exact first-order equations
Solve second-order linear equations with the characteristic equation and undetermined coefficients
Use Laplace transforms to solve initial value problems
Analyze linear systems with eigenvalues and phase portraits

What's inside your Differential Equations course

A sample path Learnr builds for Differential Equations. Your real course adapts to your goal and level.

1. What differential equations are and how they model change
2. First-order equations and slope fields
3. Applications: growth, decay, mixing and cooling
4. Second-order linear equations and oscillations
5. Laplace transforms
6. Systems of equations and phase portraits

How long does it take to learn Differential Equations?

A first course in ordinary differential equations takes most learners about 3 to 4 months at 5 to 7 hours a week, matching a typical semester. Strong integration skills and some linear algebra make the pace much smoother.

What you need before you start

Single-variable calculus through integration techniques. Some linear algebra (matrices, eigenvalues) is needed for systems, and many courses teach just enough of it along the way.

Common mistakes to avoid

Forgetting the constant of integration, or adding it too late, which breaks the fit to initial conditions.
Choosing a particular solution guess that duplicates part of the homogeneous solution, instead of multiplying the guess by t.
Learning only the solution recipes without sketching or interpreting the solution, so you miss what the model actually predicts.

How Learnr builds your course

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Differential Equations questions, answered

What is the difference between ordinary and partial differential equations?

Ordinary differential equations involve functions of one variable, such as position over time. Partial differential equations involve several variables, such as temperature across space and time, and include the heat and wave equations. Most people learn ODEs first.

Do I need linear algebra for differential equations?

For first and second-order equations, no. For systems of equations, eigenvalues and eigenvectors are central, so some linear algebra helps a great deal. Many courses teach the needed pieces alongside.