1.
Basic
Solve y' = 3y with y(0) = 2.
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Answer:
The rate is proportional to y, so the solution is exponential with rate 3.
First-order modeling practice problems
You turn a short word problem into a differential equation. You solve it with the separable and linear methods you already know. Then you answer a question such as how long something takes. Recognition cue: the problem describes a real quantity changing over time, and the rate rule is given in words. Examples: money, temperature, salt, population, speed. You will practice: Exponential Change, Approach to Equilibrium, Mixing Tanks, Logistic Growth.
2.
Basic
A chemical sample decays by N' = -0.5N. At t = 0 there are 80 mg. Find N(t).
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Answer:
Decay proportional to the amount gives a decaying exponential.
3.
Basic
A radioactive isotope has half-life 10 years and decay law N' = -kN. Find k.
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Answer:
The half-life formula gives k directly.
4.
Basic
A bacteria population follows P' = 0.04P with P(0) = 1000. At what time t does P reach 2000?
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Answer:
Write the exponential solution, then set it equal to 2000.
5.
Basic
A bank account earns continuous interest by A' = 0.05A, with A(0) = 2000 dollars. Find the balance after 20 years.
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Answer:
The balance is exponential. At t = 20 the exponent is 0.05 times 20 = 1.
6.
Basic
A cup of tea obeys T' = -0.2(T - 20) with T(0) = 100. Find T(t).
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Answer:
The gap from 20 starts at 80 and decays at rate 0.2.
7.
Basic
A cold drink at 5 degrees sits in a 25 degree room and obeys T' = -0.1(T - 25). Find T(t).
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Answer:
The gap starts at 5 - 25 = -20 and decays at rate 0.1. The drink warms toward 25.
8.
Basic
A 50 L tank is full of pure water. Brine with 3 g of salt per liter enters at 2 L/min. The mixed solution leaves at 2 L/min. Let A(t) be the grams of salt in the tank. Write the differential equation for A.
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Answer:
Salt enters at 3 times 2 = 6 g/min. Salt leaves at concentration A/50 times 2 L/min.
9.
Basic
A population follows P' = 2P(1 - P/500) with P(0) = 10. Find the limit of P(t) as t goes to infinity.
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Answer:
This is logistic with carrying capacity 500. Any positive start approaches the cap.
10.
Basic
A falling object obeys v' = 9.8 - 0.49v. Find the terminal velocity.
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Answer:
Terminal velocity makes the acceleration zero. Set v' = 0.
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More differential equations practice
- What an ODE is practice problems
- Separable equations practice problems
- First-order linear practice problems
- Exact & substitutions practice problems
- Second-order homogeneous practice problems
- Nonhomogeneous equations practice problems
- Oscillations & resonance practice problems
- Laplace transforms practice problems
- Linear systems practice problems