Recipe
Solve an initial value problem
Use when: you're given a function's derivative and one known value, and asked for the specific function itself
Goal: one particular antiderivative — a single function, not a whole family
Taught in Initial value problems
Quick recipe
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1
Antidifferentiate the given derivative, remembering the +C.
Example
Given s'(t)=6t: s(t)=3t^2+C.
Why: without C, this step only recovers one possible function out of infinitely many that share the same derivative.
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2
Plug the given input and output values into the general antiderivative from Step 1.
Example
Given s(0)=4: 3(0)^2+C=4.
Why: the initial condition is the one piece of information that distinguishes this particular function from every other member of the family.
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3
Solve the resulting equation for the constant.
Example
0+C=4, so C=4.
Why: this is the step that turns a whole family of functions into exactly one.
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4
Substitute the solved value of C back into the general antiderivative from Step 1.
Example
s(t)=3t^2+4.
Why: the answer to an initial value problem is always this one specific function, not the general family from Step 1.
Common mistakes
- Forgetting the +C when antidifferentiating, which leaves nothing for the initial condition to solve for.
- Substituting the initial condition into the ORIGINAL derivative instead of into the antiderivative.
- Solving for C correctly but then reporting the general family instead of the particular function.
- Mixing up which given number is the input and which is the output when substituting.