Training Differential Equations Placement Test Practice — Differential Equations
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Placement Test Practice — Differential Equations

24 min Differential Equations

Placement Test Practice — Differential Equations

This placement test covers the full scope of the Differential Equations module: classifying equations by order, interpreting slope fields and initial conditions, solving separable and first-order linear equations, and recognizing second-order oscillation models. These sixteen questions are modeled on the kinds of problems you would encounter on a college placement exam or introductory ODE course final.

Work each problem from memory first, then check against the answer key. Any question that gives you trouble points to a lesson worth revisiting. Repeated practice across varied problem types is the most effective way to build fluency with differential equations.

Practice Test — 16 Questions

1. What is the order of $y'''+y=0$?
2. What does $y(0)=3$ mean?
3. Solve $y'=2x$.
4. Solve $y'=4y$.
5. Solve $y'=ky$ in words.
6. What is a slope field?
7. What is the integrating factor for $y'+y=1$?
8. Write the standard form of a first-order linear equation.
9. What does separable mean?
10. What kind of behavior does $y''+\omega^2 y=0$ model?
11. In motion, what is $y'$?
12. In motion, what is $y''$?
13. Solve $y'=3y$ with $y(0)=2$.
14. Why are initial conditions used?
15. What happens when the integrating-factor method works?
16. Give one application of differential equations.
Show Answer Key

1. Third order

2. The solution passes through $(0,3)$

3. $y=x^2+C$

4. $y=Ce^{4x}$

5. Exponential growth or decay depending on the sign of $k$

6. A picture of slopes for possible solution curves

7. $e^x$

8. $y'+P(x)y=Q(x)$

9. Variables can be split and integrated on opposite sides

10. Oscillation

11. Velocity or first rate of change

12. Acceleration or second rate of change

13. $y=2e^{3x}$

14. To choose one specific solution from a family

15. The left side becomes the derivative of a product

16. Population growth, cooling, springs, circuits, and many others

Second-Order Models and Oscillation Back to Module