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)17 dm = ?

A radioactive tracer has a half-life of 23 days. A sample initially contains 200 mg. The decay follows dm/dt = −k·m. How much m remains after 17 days?

m0 = 200 mgt½ = 23 dt = 17 d
m = mg
2.Easy
rr² + br + c?

For the equation y″ + 4y′ + 3y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 4y′ + 3y = 0
r =
3.Medium
tyy(0) = 3y(2) = ?

Solve y″ + 5y′ + 4y = 0 with y(0) = 3 and y′(0) = 0. Find y(2).

y″ + 5y′ + 4y = 0y(0) = 3y′(0) = 0
y(2) =
4.Medium
t (min)T (°C)20Ta​240 °C348 °C

A forged steel part at 420 °C is left to cool in a workshop at 21 °C. After 20 min its temperature is 240 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 348 °C?

T0 = 420 °CTa = 21 °CT(20) = 240 °CT = 348 °C
t = min
5.Hard
xy13(1, 0)y(3) = ?

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

y′ + 3y/x = 5y(1) = 0
y(3) =
6.Hard
xy13(1, 1)y(3) = ?

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

y′ + y/x = 3xy(1) = 1
y(3) =

Answer key

  1. 1. m = 119.8 mg
  2. 2. r = −1
  3. 3. y(2) = 0.541
  4. 4. t = 6.635 min
  5. 5. y(3) = 3.704
  6. 6. y(3) = 9

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