Introduction to Differential Equations Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Introduction to Differential Equations.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

An equation that contains an unknown function and one or more of its derivatives. Solving a DE means finding the function(s) that satisfy the equation.

An algebraic equation like x^2 = 4 asks 'what number satisfies this?' A differential equation like \frac{dy}{dx} = 2x asks 'what function has this derivative?' The answer isn't a number but a family of functions: y = x^2 + C.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A DE relates a function to its derivatives. The order is the highest derivative that appears. The general solution contains arbitrary constants (one per order); initial conditions pin down specific solutions.

Common stuck point: Don't forget the arbitrary constant C in the general solution—it represents an entire family of curves. An initial condition y(x_0) = y_0 determines the specific solution from this family.

Sense of Study hint: Verify your solution by plugging it back into the original equation and checking that both sides are equal.

Worked Examples

Example 1

easy
Verify y = 3e^{2x} solves y' = 2y, and find the particular solution with y(0) = 5.

Solution

  1. 1
    y' = 6e^{2x} = 2(3e^{2x}) = 2y. ✓
  2. 2
    General: y = Ce^{2x}. Apply y(0)=5: C=5.
  3. 3
    Particular: y = 5e^{2x}.

Answer

Verified; particular solution y = 5e^{2x}
Verify by substitution; the initial condition pins down C.

Example 2

medium
Find the general and particular (y(0)=4) solution to y' = 3x^2+1.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Verify that y = \sin x + 2\cos x satisfies y'' + y = 0.

Example 2

medium
Find the particular solution to y' = -2y with y(0)=3.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

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