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Mathematics · lesson 1 of 3

Initial-value problems and equilibrium

Solving a differential equation produces constants; initial conditions select a particular solution.

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Learn the ideas

Prerequisites: Differentiation, exponentials and integration.

General versus particular

Solving a differential equation produces constants; initial conditions select a particular solution.

Equilibrium

A constant solution is found by setting the derivative to zero.

Existence and uniqueness

Conditions on the right-hand side matter; continuity alone does not always imply uniqueness.

Verification

Differentiate a proposed solution and check both the equation and the initial condition.

Worked examples

Example 1

Solve y′=3y with y(0)=2.

Credit a general exponential solution, applying y(0), and correct substitution.

y(t)=2e^(3t).

Example 2

For y′+2y=8, find the equilibrium solution value.

Set y′=0 so 2y=8.

4

Check your understanding

8 questions

Decimals or fractions are accepted for numeric answers. Proofs and written responses use sample answers and require human marking. This is practice, not a protected examination.

1 Solve y′=3y with y(0)=2.

Sample answer and marking guidance

y(t)=2e^(3t).

Credit a general exponential solution, applying y(0), and correct substitution.

2 For y′+2y=8, find the equilibrium solution value.

Show answer and explanation

4

Set y′=0 so 2y=8.

3 Solve y′+2y=8 with y(0)=1.

Sample answer and marking guidance

y(t)=4−3e^(−2t).

Credit the equilibrium plus a decaying homogeneous solution and the initial condition.

4 For y′=−5y, what happens to small deviations from y=0 as t increases?

Show answer and explanation

They decay exponentially

The solution is Ce^(−5t), so the equilibrium is asymptotically stable.

5 For y′=2t and y(0)=3, calculate y(2).

Show answer and explanation

7

Integration gives y=t²+3, so y(2)=7.

6 Why does checking only the initial condition fail to verify a proposed solution?

Sample answer and marking guidance

Infinitely many functions pass through the initial point. The function must also satisfy the differential equation throughout its interval.

Credit the distinction between a pointwise constraint and a differential relation.

7 Which form is the homogeneous part of y′+ay=b for a>0?

Show answer and explanation

Ce^(−at)

The homogeneous equation is y′+ay=0.

8 Explain why dividing by y when solving y′=y can hide a solution.

Sample answer and marking guidance

Division assumes y≠0. The constant solution y=0 must be checked separately, though it is recovered by allowing C=0 in the final exponential family.

Credit the excluded case and direct verification.

Further reading and review references

These are reference links, not copied exercises, institutional endorsement or proof of independent review.