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
rr² + br + c?

For the equation y″ + 10y′ + 24y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 10y′ + 24y = 0
r =
3.Medium
t (min)T (°C)5Ta​61 °C

A cup of coffee at 82 °C is left in a room at 18 °C. After 5 min its temperature is 61 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 82 °CTa = 18 °CT(5) = 61 °C
k = min⁻¹
4.Medium
tyy(0) = 5y(0.5) = ?

Solve y″ + 8y′ + 15y = 0 with y(0) = 5 and y′(0) = 0. Find y(0.5).

y″ + 8y′ + 15y = 0y(0) = 5y′(0) = 0
y(0.5) =
5.Hard
t (min)m (kg)90 minm = ?

A tank holds 400 L of pure water. Brine with 0.25 kg of salt per litre flows in at 5 L/min, and the well-mixed solution flows out at the same rate. How much salt m is in the tank after 90 min?

V = 400 Lm0 = 0 kgc = 0.25 kg/Lr = 5 L/min
m = kg
6.Hard
xy12(1, 5)y(2) = ?

Solve the initial value problem y′ + 3y/x = 4x2 for x > 0, with y(1) = 5. Find y(2).

y′ + 3y/x = 4x2y(1) = 5
y(2) =

Answer key

  1. 1. r = −1
  2. 2. r = −4
  3. 3. k = 0.07954 min⁻¹
  4. 4. y(0.5) = 2.173
  5. 5. m = 67.53 kg
  6. 6. y(2) = 5.875

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