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)6 dm = ?

A radioactive tracer has a half-life of 21 days. A sample initially contains 120 mg. The decay follows dm/dt = −k·m. How much m remains after 6 days?

m0 = 120 mgt½ = 21 dt = 6 d
m = mg
2.Easy
t (h)y2 h800

A quantity grows according to dy/dt = k·y with k > 0. It is 200 at t = 0 and 800 at t = 2 h. Find the rate constant k in h⁻¹.

y(0) = 200y(2) = 800
k = h⁻¹
3.Medium
tyy(0) = 5y(1) = ?

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

y″ + 4y′ + 4y = 0y(0) = 5y′(0) = 0
y(1) =
4.Medium
t (min)T (°C)14Ta​68 °C42 °C

A cup of coffee at 93 °C is left in a room at 20 °C. After 14 min its temperature is 68 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 42 °C?

T0 = 93 °CTa = 20 °CT(14) = 68 °CT = 42 °C
t = min
5.Hard
ty1y(1) = ?

Solve y′ + 3y = 5e−t with y(0) = 4. Find y(1).

y′ + 3y = 5e−ty(0) = 4
y(1) =
6.Hard
xy12(1, −2)y(2) = ?

Solve the initial value problem y′ + y/x = 6x2 for x > 0, with y(1) = −2. Find y(2).

y′ + y/x = 6x2y(1) = −2
y(2) =

Answer key

  1. 1. m = 98.44 mg
  2. 2. k = 0.6931 h⁻¹
  3. 3. y(1) = 2.03
  4. 4. t = 40.05 min
  5. 5. y(1) = 0.9944
  6. 6. y(2) = 10.25

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