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
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 =
2.Easy
t (days)m (mg)30 dm = ?

A radioactive tracer has a half-life of 8 days. A sample initially contains 120 mg. The decay follows dm/dt = −k·m. How much m remains after 30 days?

m0 = 120 mgt½ = 8 dt = 30 d
m = mg
3.Medium
tyy(0) = 1y(0.5) = ?

Solve y″ + 8y′ + 15y = 0 with y(0) = 1 and y′(0) = 6. Find y(0.5).

y″ + 8y′ + 15y = 0y(0) = 1y′(0) = 6
y(0.5) =
4.Medium
t (min)T (°C)16Ta​37 °C27 °C

A cup of coffee at 95 °C is left in a room at 19 °C. After 16 min its temperature is 37 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 27 °C?

T0 = 95 °CTa = 19 °CT(16) = 37 °CT = 27 °C
t = min
5.Hard
ty1y(1) = ?

Solve y′ + 0.5y = 8e−3t with y(0) = 1. Find y(1).

y′ + 0.5y = 8e−3ty(0) = 1
y(1) =
6.Hard
xy13(1, −3)y(3) = ?

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

y′ + y/x = 2x2y(1) = −3
y(3) =

Answer key

  1. 1. r = −1
  2. 2. m = 8.919 mg
  3. 3. y(0.5) = 0.8578
  4. 4. t = 25.01 min
  5. 5. y(1) = 2.388
  6. 6. y(3) = 12.33

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