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)y8 h160

A quantity decays according to dy/dt = −k·y with k > 0. It is 400 at t = 0 and 160 at t = 8 h. Find the rate constant k in h⁻¹.

y(0) = 400y(8) = 160
k = h⁻¹
2.Easy
t (days)m (mg)90 dm = ?

A radioactive tracer has a half-life of 25 days. A sample initially contains 80 mg. The decay follows dm/dt = −k·m. How much m remains after 90 days?

m0 = 80 mgt½ = 25 dt = 90 d
m = mg
3.Medium
tyy(0) = 3y(1.5) = ?

Solve y″ + 8y′ + 16y = 0 with y(0) = 3 and y′(0) = 0. Find y(1.5).

y″ + 8y′ + 16y = 0y(0) = 3y′(0) = 0
y(1.5) =
4.Medium
t (min)T (°C)25Ta​30 °C47 °C

A cup of coffee at 86 °C is left in a room at 21 °C. After 25 min its temperature is 30 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 47 °C?

T0 = 86 °CTa = 21 °CT(25) = 30 °CT = 47 °C
t = min
5.Hard
xy13(1, 1)y(3) = ?

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

y′ + 3y/x = 5xy(1) = 1
y(3) =
6.Hard
tyy(0) = 5y(1) = ?

Solve y″ + 4y′ + 29y = 0 with y(0) = 5 and y′(0) = 0 (released from rest). Find y(1); angles in radians.

y″ + 4y′ + 29y = 0y(0) = 5y′(0) = 0
y(1) =

Answer key

  1. 1. k = 0.1145 h⁻¹
  2. 2. m = 6.598 mg
  3. 3. y(1.5) = 0.05205
  4. 4. t = 11.59 min
  5. 5. y(3) = 9
  6. 6. y(1) = −0.06761

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