Differential equations

Model change with initial or boundary conditions and verify that solutions satisfy them.

Meaning

A differential equation relates an unknown function to its derivatives. Ordinary differential equations involve one independent variable; partial differential equations can describe changes in space and time together. Both are used to model natural phenomena.

Example

The solution of with is Differentiating verifies the equation, and setting gives the initial value 3, so both conditions hold.For y′−2y=0, differentiating ye⁻²ᵗ gives e⁻²ᵗ(y′−2y)=0. Thus the product is a constant C and y=Ce²ᵗ. The initial condition y(0)=3 fixes C=3.A differential equation alone may not specify a unique solution. Check initial or boundary conditions and the conditions for existence and uniqueness.

Practice for this concept

Scope and references

Topics

Ordinary and partial differential equations, models of changing phenomena

Learning goal

Model change with initial or boundary conditions and verify that solutions satisfy them.

Helpful background

Explore the functions, calculus, and topic-specific linear algebra needed for the chosen topic.

Checking differential-equation solutions and conditions

These tasks do not assess general existence and uniqueness or other initial and boundary-value problems.

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