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″ + 8y′ + 15y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 8y′ + 15y = 0
r =
2.Easy
ty1

A mass–spring–damper is described by y″ + 5.6y′ + 16y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 5.6y′ + 16y = 0
ζ =
3.Medium
tyy(0) = 2y(0.5) = ?

Solve y″ + 6y′ + 5y = 0 with y(0) = 2 and y′(0) = 0. Find y(0.5).

y″ + 6y′ + 5y = 0y(0) = 2y′(0) = 0
y(0.5) =
4.Medium
t (min)T (°C)49Ta​229 °C

A forged steel part at 360 °C is left to cool in a workshop at 21 °C. After 49 min its temperature is 229 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 360 °CTa = 21 °CT(49) = 229 °C
k = min⁻¹
5.Hard
tyy(0) = 1y(0.75) = ?

Solve y″ + y′ + 25.25y = 0 with y(0) = 1 and y′(0) = 0 (released from rest). Find y(0.75); angles in radians.

y″ + y′ + 25.25y = 0y(0) = 1y′(0) = 0
y(0.75) =
6.Hard
xy12(1, 3)y(2) = ?

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

y′ + y/x = 4xy(1) = 3
y(2) =

Answer key

  1. 1. r = −3
  2. 2. ζ = 0.7
  3. 3. y(0.5) = 1.475
  4. 4. k = 0.009969 min⁻¹
  5. 5. y(0.75) = −0.6032
  6. 6. y(2) = 6.167

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