Differential equations worksheet with answers

Free printable Differential equations worksheet: 6 problems from easy to exam level, with an answer key. Print it or save it as a PDF.

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1.Easy
t (h)y5 h325

A quantity decays according to dy/dt = −k·y with k > 0. It is 500 at t = 0 and 325 at t = 5 h. Find the rate constant k in h⁻¹.

y(0) = 500y(5) = 325
k = h⁻¹
2.Easy
t (days)m (mg)90 dm = ?

A radioactive tracer has a half-life of 29 days. A sample initially contains 120 mg. The decay follows dm/dt = −k·m. How much m remains after 90 days?

m0 = 120 mgt½ = 29 dt = 90 d
m = mg
3.Medium
t (min)T (°C)49Ta​75 °C228 °C

A forged steel part at 380 °C is left to cool in a workshop at 25 °C. After 49 min its temperature is 75 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 228 °C?

T0 = 380 °CTa = 25 °CT(49) = 75 °CT = 228 °C
t = min
4.Medium
tyy(0) = 2y(1.5) = ?

Solve y″ + 8y′ + 16y = 0 with y(0) = 2 and y′(0) = -1. Find y(1.5).

y″ + 8y′ + 16y = 0y(0) = 2y′(0) = -1
y(1.5) =
5.Hard
xy12(1, −3)y(2) = ?

Solve the initial value problem y′ + 3y/x = 5x for x > 0, with y(1) = −3. Find y(2).

y′ + 3y/x = 5xy(1) = −3
y(2) =
6.Hard
ty1.5y(1.5) = ?

Solve y′ + 2y = 10e−0.5t with y(0) = 1. Find y(1.5).

y′ + 2y = 10e−0.5ty(0) = 1
y(1.5) =

Answer key

  1. 1. k = 0.08616 h⁻¹
  2. 2. m = 13.96 mg
  3. 3. t = 13.97 min
  4. 4. y(1.5) = 0.03098
  5. 5. y(2) = 3.5
  6. 6. y(1.5) = 2.867

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