solution rises
↝CALCULUS II STUDY GUIDE
First-Order Differential Equations
Solve separable equations, read slope fields, and model growth, decay, and logistic behavior.
8 key ideas8 flashcards5 practice questions
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solution is locally flat
solution falls
Key ideas to know
Start with the relationships between ideas. Then close the notes and explain each one from memory.
- 01
A differential equation connects an unknown function with one or more derivatives.
- 02
An initial condition selects one solution from a family.
- 03
A slope field records the derivative at many points without solving explicitly.
- 04
A separable equation can place y terms with dy and x terms with dx.
- 05
After integration, an implicit solution may be sufficient or may be solved for y.
- 06
Exponential models have rate proportional to the current amount.
- 07
A logistic model slows as the amount approaches its carrying capacity.
- 08
A proposed solution is checked by substitution into both the equation and initial condition.
PART OF THE CALCULUS II GUIDE
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Open the full Calculus II guide →Two ideas worth correcting now
NOT QUITE
→A slope field contains only one solution
USE THIS INSTEAD
Many solution curves can follow the same field from different starting points.
NOT QUITE
→Separating variables can never lose an answer
USE THIS INSTEAD
Division by an expression containing y can remove an equilibrium solution.
Flashcards
Answer before opening each card. The effort to retrieve is part of the learning.
1What does an initial condition do?Show answer ＋
It determines the constant for a particular solution.
2What does a slope-field segment show?Show answer ＋
The derivative at its point.
3What makes an equation separable?Show answer ＋
Its variables can be placed on opposite sides with their differentials.
4What equation models proportional growth?Show answer ＋
dy/dt = ky.
5What is the solution of dy/dt = ky?Show answer ＋
y = Ceᵏᵗ.
6What does k<0 mean in the exponential model?Show answer ＋
Exponential decay.
7What does logistic carrying capacity mean?Show answer ＋
The limiting amount K in dy/dt = ry(1−y/K).
8How is a solution checked?Show answer ＋
Differentiate it, substitute into the equation, and test the initial condition.
Explain it in your own words
Use the answer as a check after you have written or spoken your response.
01Solve dy/dx = 2xy with y(0)=3.
Separation gives ln|y|=x²+C, so y=3eˣ².
02What can a slope field tell before solving?
Where solutions rise, fall, flatten, or approach equilibrium curves.
03Why can dividing by a y-expression lose solutions?
A zero of that expression may be a constant equilibrium solution and must be checked separately.
04How does logistic growth differ from exponential growth?
Its per-capita growth falls as the amount approaches carrying capacity.
05Why must units be attached to a rate constant?
They make the derivative equation dimensionally consistent and clarify the time scale.
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