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 (days)m (mg)28 dm = ?

A radioactive tracer has a half-life of 11 days. A sample initially contains 200 mg. The decay follows dm/dt = −k·m. How much m remains after 28 days?

m0 = 200 mgt½ = 11 dt = 28 d
m = mg
2.Easy
rr² + br + c?

For the equation y″ + 7y′ + 10y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 7y′ + 10y = 0
r =
3.Medium
tyy(0) = 4y(1) = ?

Solve y″ + 5y′ + 6y = 0 with y(0) = 4 and y′(0) = 0. Find y(1).

y″ + 5y′ + 6y = 0y(0) = 4y′(0) = 0
y(1) =
4.Medium
t (min)T (°C)18Ta​339 °C382 °C

A forged steel part at 400 °C is left to cool in a workshop at 29 °C. After 18 min its temperature is 339 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 382 °C?

T0 = 400 °CTa = 29 °CT(18) = 339 °CT = 382 °C
t = min
5.Hard
xy13(1, 0)y(3) = ?

Solve the initial value problem y′ + 2y/x = 3x2 for x > 0, with y(1) = 0. Find y(3).

y′ + 2y/x = 3x2y(1) = 0
y(3) =
6.Hard
xy12(1, −3)y(2) = ?

Solve the initial value problem y′ + y/x = 6 for x > 0, with y(1) = −3. Find y(2).

y′ + y/x = 6y(1) = −3
y(2) =

Answer key

  1. 1. m = 34.26 mg
  2. 2. r = −2
  3. 3. y(1) = 1.226
  4. 4. t = 4.984 min
  5. 5. y(3) = 16.13
  6. 6. y(2) = 3

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