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Second-order linear equations practice problems

Constant coefficients: the characteristic equation and its three cases. 70 problems with worked solutions; in 70 of them every equation is proved by a computer algebra system.

10 worked problems

Try each one before opening the solution. Every line is labeled with what checked it. How we verify

Problem 6.37

hard
Solve \( \displaystyle y'' + 2y' + 10y = 0 \) with \( \displaystyle y(0) = 1,\ y'(0) = 3 \).
  1. Try y = e^(mx): the characteristic equation is m² + bm + c = 0.
    Reviewed
  2. \[ \left(m + 1 - 3 i\right) \left(m + 1 + 3 i\right) = m^{2} + 2 m + 10 \]
    Roots: -1 - 3*I, -1 + 3*I.✓ Proved
  3. Roots α ± βi give y = e^(αx)(C₁cos βx + C₂sin βx).
    Reviewed
  4. Use y(0) and y'(0) to find C₁ and C₂.
    Reviewed
  5. \[ \left(\frac{40 \sin{\left(3 x \right)}}{3} + 10 \cos{\left(3 x \right)}\right) e^{- x} + 2 \frac{d}{d x} \left(\frac{4 \sin{\left(3 x \right)}}{3} + \cos{\left(3 x \right)}\right) e^{- x} + \frac{d^{2}}{d x^{2}} \left(\frac{4 \sin{\left(3 x \right)}}{3} + \cos{\left(3 x \right)}\right) e^{- x} = 0 \]
    The solution satisfies the equation.✓ Proved
  6. \[ 1 \]
    y(0).Reviewed
  7. \[ 3 \]
    y'(0).Reviewed
Answer \( y = \left(\frac{4 \sin{\left(3 x \right)}}{3} + \cos{\left(3 x \right)}\right) e^{- x} \)

✓ Nihil obstat Lines: 2 proved, 5 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 6.108

hard
Solve \( \displaystyle y'' - 6y' + 9y = 0 \) with \( \displaystyle y(0) = -3,\ y'(0) = -2 \).
  1. Try y = e^(mx): the characteristic equation is m² + bm + c = 0.
    Reviewed
  2. \[ \left(m - 3\right)^{2} = m^{2} - 6 m + 9 \]
    Roots: 3.✓ Proved
  3. A repeated root r gives y = (C₁ + C₂x)e^(rx).
    Reviewed
  4. Use y(0) and y'(0) to find C₁ and C₂.
    Reviewed
  5. \[ \left(63 x - 27\right) e^{3 x} - 6 \frac{d}{d x} \left(7 x - 3\right) e^{3 x} + \frac{d^{2}}{d x^{2}} \left(7 x - 3\right) e^{3 x} = 0 \]
    The solution satisfies the equation.✓ Proved
  6. \[ -3 \]
    y(0).Reviewed
  7. \[ -2 \]
    y'(0).Reviewed
Answer \( y = \left(7 x - 3\right) e^{3 x} \)

✓ Nihil obstat Lines: 2 proved, 5 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 6.112

hard
Solve \( \displaystyle y'' + 1y' - 6y = 0 \) with \( \displaystyle y(0) = 1,\ y'(0) = -2 \).
  1. Try y = e^(mx): the characteristic equation is m² + bm + c = 0.
    Reviewed
  2. \[ \left(m - 2\right) \left(m + 3\right) = m^{2} + m - 6 \]
    Roots: 2, -3.✓ Proved
  3. Two real roots give y = C₁e^(r₁x) + C₂e^(r₂x).
    Reviewed
  4. Use y(0) and y'(0) to find C₁ and C₂.
    Reviewed
  5. \[ \frac{\left(- 6 e^{5 x} - 24\right) e^{- 3 x}}{5} + \frac{d}{d x} \frac{\left(e^{5 x} + 4\right) e^{- 3 x}}{5} + \frac{d^{2}}{d x^{2}} \frac{\left(e^{5 x} + 4\right) e^{- 3 x}}{5} = 0 \]
    The solution satisfies the equation.✓ Proved
  6. \[ 1 \]
    y(0).Reviewed
  7. \[ -2 \]
    y'(0).Reviewed
Answer \( y = \frac{\left(e^{5 x} + 4\right) e^{- 3 x}}{5} \)

✓ Nihil obstat Lines: 2 proved, 5 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 6.136

hard
Solve \( \displaystyle y'' + 6y' + 9y = 0 \) with \( \displaystyle y(0) = -2,\ y'(0) = 3 \).
  1. Try y = e^(mx): the characteristic equation is m² + bm + c = 0.
    Reviewed
  2. \[ \left(m + 3\right)^{2} = m^{2} + 6 m + 9 \]
    Roots: -3.✓ Proved
  3. A repeated root r gives y = (C₁ + C₂x)e^(rx).
    Reviewed
  4. Use y(0) and y'(0) to find C₁ and C₂.
    Reviewed
  5. \[ \left(- 27 x - 18\right) e^{- 3 x} + 6 \frac{d}{d x} \left(- 3 x - 2\right) e^{- 3 x} + \frac{d^{2}}{d x^{2}} \left(- 3 x - 2\right) e^{- 3 x} = 0 \]
    The solution satisfies the equation.✓ Proved
  6. \[ -2 \]
    y(0).Reviewed
  7. \[ 3 \]
    y'(0).Reviewed
Answer \( y = \left(- 3 x - 2\right) e^{- 3 x} \)

✓ Nihil obstat Lines: 2 proved, 5 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 6.138

hard
Solve \( \displaystyle y'' + 6y' + 9y = 0 \) with \( \displaystyle y(0) = -1,\ y'(0) = -2 \).
  1. Try y = e^(mx): the characteristic equation is m² + bm + c = 0.
    Reviewed
  2. \[ \left(m + 3\right)^{2} = m^{2} + 6 m + 9 \]
    Roots: -3.✓ Proved
  3. A repeated root r gives y = (C₁ + C₂x)e^(rx).
    Reviewed
  4. Use y(0) and y'(0) to find C₁ and C₂.
    Reviewed
  5. \[ \left(- 45 x - 9\right) e^{- 3 x} + 6 \frac{d}{d x} \left(- 5 x - 1\right) e^{- 3 x} + \frac{d^{2}}{d x^{2}} \left(- 5 x - 1\right) e^{- 3 x} = 0 \]
    The solution satisfies the equation.✓ Proved
  6. \[ -1 \]
    y(0).Reviewed
  7. \[ -2 \]
    y'(0).Reviewed
Answer \( y = \left(- 5 x - 1\right) e^{- 3 x} \)

✓ Nihil obstat Lines: 2 proved, 5 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 6.156

hard
Solve \( \displaystyle y'' + 2y' + 5y = 0 \) with \( \displaystyle y(0) = -2,\ y'(0) = 0 \).
  1. Try y = e^(mx): the characteristic equation is m² + bm + c = 0.
    Reviewed
  2. \[ \left(m + 1 - 2 i\right) \left(m + 1 + 2 i\right) = m^{2} + 2 m + 5 \]
    Roots: -1 - 2*I, -1 + 2*I.✓ Proved
  3. Roots α ± βi give y = e^(αx)(C₁cos βx + C₂sin βx).
    Reviewed
  4. Use y(0) and y'(0) to find C₁ and C₂.
    Reviewed
  5. \[ \left(- 5 \sin{\left(2 x \right)} - 10 \cos{\left(2 x \right)}\right) e^{- x} + 2 \frac{d}{d x} \left(- \sin{\left(2 x \right)} - 2 \cos{\left(2 x \right)}\right) e^{- x} + \frac{d^{2}}{d x^{2}} \left(- \sin{\left(2 x \right)} - 2 \cos{\left(2 x \right)}\right) e^{- x} = 0 \]
    The solution satisfies the equation.✓ Proved
  6. \[ -2 \]
    y(0).Reviewed
  7. \[ 0 \]
    y'(0).Reviewed
Answer \( y = - \left(\sin{\left(2 x \right)} + 2 \cos{\left(2 x \right)}\right) e^{- x} \)

✓ Nihil obstat Lines: 2 proved, 5 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 6.181

hard
Solve \( \displaystyle y'' - 4y' + 4y = 0 \) with \( \displaystyle y(0) = -3,\ y'(0) = 3 \).
  1. Try y = e^(mx): the characteristic equation is m² + bm + c = 0.
    Reviewed
  2. \[ \left(m - 2\right)^{2} = m^{2} - 4 m + 4 \]
    Roots: 2.✓ Proved
  3. A repeated root r gives y = (C₁ + C₂x)e^(rx).
    Reviewed
  4. Use y(0) and y'(0) to find C₁ and C₂.
    Reviewed
  5. \[ \left(36 x - 12\right) e^{2 x} - 4 \frac{d}{d x} \left(9 x - 3\right) e^{2 x} + \frac{d^{2}}{d x^{2}} \left(9 x - 3\right) e^{2 x} = 0 \]
    The solution satisfies the equation.✓ Proved
  6. \[ -3 \]
    y(0).Reviewed
  7. \[ 3 \]
    y'(0).Reviewed
Answer \( y = \left(9 x - 3\right) e^{2 x} \)

✓ Nihil obstat Lines: 2 proved, 5 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 6.208

hard
Solve \( \displaystyle y'' + 4y' + 5y = 0 \) with \( \displaystyle y(0) = -1,\ y'(0) = -2 \).
  1. Try y = e^(mx): the characteristic equation is m² + bm + c = 0.
    Reviewed
  2. \[ \left(m + 2 - i\right) \left(m + 2 + i\right) = m^{2} + 4 m + 5 \]
    Roots: -2 - I, -2 + I.✓ Proved
  3. Roots α ± βi give y = e^(αx)(C₁cos βx + C₂sin βx).
    Reviewed
  4. Use y(0) and y'(0) to find C₁ and C₂.
    Reviewed
  5. \[ \left(- 20 \sin{\left(x \right)} - 5 \cos{\left(x \right)}\right) e^{- 2 x} + 4 \frac{d}{d x} \left(- 4 \sin{\left(x \right)} - \cos{\left(x \right)}\right) e^{- 2 x} + \frac{d^{2}}{d x^{2}} \left(- 4 \sin{\left(x \right)} - \cos{\left(x \right)}\right) e^{- 2 x} = 0 \]
    The solution satisfies the equation.✓ Proved
  6. \[ -1 \]
    y(0).Reviewed
  7. \[ -2 \]
    y'(0).Reviewed
Answer \( y = - \left(4 \sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{- 2 x} \)

✓ Nihil obstat Lines: 2 proved, 5 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 6.209

hard
Solve \( \displaystyle y'' + 2y' + 2y = 0 \) with \( \displaystyle y(0) = 3,\ y'(0) = -2 \).
  1. Try y = e^(mx): the characteristic equation is m² + bm + c = 0.
    Reviewed
  2. \[ \left(m + 1 - i\right) \left(m + 1 + i\right) = m^{2} + 2 m + 2 \]
    Roots: -1 - I, -1 + I.✓ Proved
  3. Roots α ± βi give y = e^(αx)(C₁cos βx + C₂sin βx).
    Reviewed
  4. Use y(0) and y'(0) to find C₁ and C₂.
    Reviewed
  5. \[ \left(2 \sin{\left(x \right)} + 6 \cos{\left(x \right)}\right) e^{- x} + 2 \frac{d}{d x} \left(\sin{\left(x \right)} + 3 \cos{\left(x \right)}\right) e^{- x} + \frac{d^{2}}{d x^{2}} \left(\sin{\left(x \right)} + 3 \cos{\left(x \right)}\right) e^{- x} = 0 \]
    The solution satisfies the equation.✓ Proved
  6. \[ 3 \]
    y(0).Reviewed
  7. \[ -2 \]
    y'(0).Reviewed
Answer \( y = \left(\sin{\left(x \right)} + 3 \cos{\left(x \right)}\right) e^{- x} \)

✓ Nihil obstat Lines: 2 proved, 5 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 6.180

medium
Solve \( \displaystyle y'' + 0y' - 1y = 0 \) with \( \displaystyle y(0) = 1,\ y'(0) = 3 \).
  1. Try y = e^(mx): the characteristic equation is m² + bm + c = 0.
    Reviewed
  2. \[ \left(m - 1\right) \left(m + 1\right) = m^{2} - 1 \]
    Roots: -1, 1.✓ Proved
  3. Two real roots give y = C₁e^(r₁x) + C₂e^(r₂x).
    Reviewed
  4. Use y(0) and y'(0) to find C₁ and C₂.
    Reviewed
  5. \[ - 3 \sinh{\left(x \right)} - \cosh{\left(x \right)} + \frac{d^{2}}{d x^{2}} \left(3 \sinh{\left(x \right)} + \cosh{\left(x \right)}\right) = 0 \]
    The solution satisfies the equation.✓ Proved
  6. \[ 1 \]
    y(0).Reviewed
  7. \[ 3 \]
    y'(0).Reviewed
Answer \( y = 3 \sinh{\left(x \right)} + \cosh{\left(x \right)} \)

✓ Nihil obstat Lines: 2 proved, 5 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

More second-order linear equations practice

Solve \( \displaystyle y'' - 2y' + 1y = 0 \) with \( \displaystyle y(0) = 1,\ y'(0) = -2 \).
Problem 6.36medium✓ Every equation proved
Solve \( \displaystyle y'' - 6y' + 9y = 0 \) with \( \displaystyle y(0) = 1,\ y'(0) = 2 \).
Problem 6.41medium✓ Every equation proved
Solve \( \displaystyle y'' + 0y' + 9y = 0 \) with \( \displaystyle y(0) = 1,\ y'(0) = 0 \).
Problem 6.78medium✓ Every equation proved
Solve \( \displaystyle y'' + 4y' + 4y = 0 \) with \( \displaystyle y(0) = 1,\ y'(0) = -3 \).
Problem 6.79medium✓ Every equation proved
Solve \( \displaystyle y'' - 4y' + 3y = 0 \) with \( \displaystyle y(0) = 2,\ y'(0) = 2 \).
Problem 6.87medium✓ Every equation proved
Solve \( \displaystyle y'' - 2y' + 1y = 0 \) with \( \displaystyle y(0) = 0,\ y'(0) = -3 \).
Problem 6.110medium✓ Every equation proved
Solve \( \displaystyle y'' - 2y' + 1y = 0 \) with \( \displaystyle y(0) = 3,\ y'(0) = 0 \).
Problem 6.132medium✓ Every equation proved
Solve \( \displaystyle y'' + 0y' + 1y = 0 \) with \( \displaystyle y(0) = -3,\ y'(0) = 0 \).
Problem 6.133medium✓ Every equation proved
Solve \( \displaystyle y'' + 4y' + 4y = 0 \) with \( \displaystyle y(0) = 0,\ y'(0) = 1 \).
Problem 6.135medium✓ Every equation proved
Solve \( \displaystyle y'' - 2y' + 1y = 0 \) with \( \displaystyle y(0) = 3,\ y'(0) = 3 \).
Problem 6.184medium✓ Every equation proved
Solve \( \displaystyle y'' - 2y' + 1y = 0 \) with \( \displaystyle y(0) = 1,\ y'(0) = -1 \).
Problem 6.185medium✓ Every equation proved
Solve \( \displaystyle y'' - 1y' - 6y = 0 \) with \( \displaystyle y(0) = -1,\ y'(0) = 2 \).
Problem 6.186medium✓ Every equation proved
Solve \( \displaystyle y'' + 6y' + 9y = 0 \) with \( \displaystyle y(0) = 1,\ y'(0) = 2 \).
Problem 6.31hard✓ Every equation proved
Solve \( \displaystyle y'' - 6y' + 9y = 0 \) with \( \displaystyle y(0) = -2,\ y'(0) = -2 \).
Problem 6.32hard✓ Every equation proved
Solve \( \displaystyle y'' - 4y' + 5y = 0 \) with \( \displaystyle y(0) = -1,\ y'(0) = -1 \).
Problem 6.33hard✓ Every equation proved
Solve \( \displaystyle y'' - 1y' - 2y = 0 \) with \( \displaystyle y(0) = 2,\ y'(0) = -1 \).
Problem 6.34hard✓ Every equation proved
Solve \( \displaystyle y'' + 5y' + 6y = 0 \) with \( \displaystyle y(0) = -2,\ y'(0) = 0 \).
Problem 6.35hard✓ Every equation proved
Solve \( \displaystyle y'' - 4y' + 5y = 0 \) with \( \displaystyle y(0) = -1,\ y'(0) = 1 \).
Problem 6.38hard✓ Every equation proved
Solve \( \displaystyle y'' + 2y' + 2y = 0 \) with \( \displaystyle y(0) = -3,\ y'(0) = 3 \).
Problem 6.39hard✓ Every equation proved
Solve \( \displaystyle y'' + 3y' + 2y = 0 \) with \( \displaystyle y(0) = -1,\ y'(0) = 0 \).
Problem 6.40hard✓ Every equation proved
Solve \( \displaystyle y'' + 2y' - 3y = 0 \) with \( \displaystyle y(0) = 1,\ y'(0) = 3 \).
Problem 6.42hard✓ Every equation proved
Solve \( \displaystyle y'' - 6y' + 9y = 0 \) with \( \displaystyle y(0) = 0,\ y'(0) = -2 \).
Problem 6.43hard✓ Every equation proved
Solve \( \displaystyle y'' - 6y' + 9y = 0 \) with \( \displaystyle y(0) = -3,\ y'(0) = -3 \).
Problem 6.44hard✓ Every equation proved
Solve \( \displaystyle y'' + 2y' + 5y = 0 \) with \( \displaystyle y(0) = 1,\ y'(0) = 3 \).
Problem 6.45hard✓ Every equation proved
Solve \( \displaystyle y'' - 2y' + 5y = 0 \) with \( \displaystyle y(0) = 2,\ y'(0) = -3 \).
Problem 6.76hard✓ Every equation proved
Solve \( \displaystyle y'' + 1y' - 2y = 0 \) with \( \displaystyle y(0) = 0,\ y'(0) = 3 \).
Problem 6.77hard✓ Every equation proved
Solve \( \displaystyle y'' + 4y' + 4y = 0 \) with \( \displaystyle y(0) = -3,\ y'(0) = -3 \).
Problem 6.80hard✓ Every equation proved
Solve \( \displaystyle y'' + 4y' + 4y = 0 \) with \( \displaystyle y(0) = 2,\ y'(0) = 2 \).
Problem 6.81hard✓ Every equation proved
Solve \( \displaystyle y'' + 2y' + 2y = 0 \) with \( \displaystyle y(0) = -2,\ y'(0) = -1 \).
Problem 6.82hard✓ Every equation proved
Solve \( \displaystyle y'' + 1y' - 6y = 0 \) with \( \displaystyle y(0) = 0,\ y'(0) = 3 \).
Problem 6.83hard✓ Every equation proved
Solve \( \displaystyle y'' - 2y' + 5y = 0 \) with \( \displaystyle y(0) = -2,\ y'(0) = 1 \).
Problem 6.84hard✓ Every equation proved
Solve \( \displaystyle y'' + 4y' + 4y = 0 \) with \( \displaystyle y(0) = 2,\ y'(0) = -2 \).
Problem 6.85hard✓ Every equation proved
Solve \( \displaystyle y'' + 5y' + 6y = 0 \) with \( \displaystyle y(0) = -2,\ y'(0) = -3 \).
Problem 6.86hard✓ Every equation proved
Solve \( \displaystyle y'' - 1y' - 6y = 0 \) with \( \displaystyle y(0) = 1,\ y'(0) = -3 \).
Problem 6.88hard✓ Every equation proved
Solve \( \displaystyle y'' - 5y' + 6y = 0 \) with \( \displaystyle y(0) = 1,\ y'(0) = 0 \).
Problem 6.89hard✓ Every equation proved
Solve \( \displaystyle y'' + 0y' - 4y = 0 \) with \( \displaystyle y(0) = 2,\ y'(0) = -1 \).
Problem 6.90hard✓ Every equation proved
Solve \( \displaystyle y'' - 4y' + 5y = 0 \) with \( \displaystyle y(0) = 3,\ y'(0) = 0 \).
Problem 6.107hard✓ Every equation proved
Solve \( \displaystyle y'' + 4y' + 4y = 0 \) with \( \displaystyle y(0) = -2,\ y'(0) = -2 \).
Problem 6.109hard✓ Every equation proved
Solve \( \displaystyle y'' + 0y' + 4y = 0 \) with \( \displaystyle y(0) = -1,\ y'(0) = -3 \).
Problem 6.111hard✓ Every equation proved
Solve \( \displaystyle y'' + 6y' + 9y = 0 \) with \( \displaystyle y(0) = -1,\ y'(0) = -1 \).
Problem 6.113hard✓ Every equation proved
Solve \( \displaystyle y'' - 6y' + 9y = 0 \) with \( \displaystyle y(0) = -2,\ y'(0) = 3 \).
Problem 6.114hard✓ Every equation proved
Solve \( \displaystyle y'' + 6y' + 9y = 0 \) with \( \displaystyle y(0) = -3,\ y'(0) = 0 \).
Problem 6.131hard✓ Every equation proved
Solve \( \displaystyle y'' + 2y' + 10y = 0 \) with \( \displaystyle y(0) = -3,\ y'(0) = 1 \).
Problem 6.134hard✓ Every equation proved
Solve \( \displaystyle y'' + 2y' + 10y = 0 \) with \( \displaystyle y(0) = 1,\ y'(0) = 1 \).
Problem 6.137hard✓ Every equation proved
Solve \( \displaystyle y'' + 4y' + 5y = 0 \) with \( \displaystyle y(0) = 3,\ y'(0) = -1 \).
Problem 6.155hard✓ Every equation proved
Solve \( \displaystyle y'' - 1y' - 2y = 0 \) with \( \displaystyle y(0) = -2,\ y'(0) = 0 \).
Problem 6.157hard✓ Every equation proved
Solve \( \displaystyle y'' - 4y' + 8y = 0 \) with \( \displaystyle y(0) = 2,\ y'(0) = 1 \).
Problem 6.158hard✓ Every equation proved
Solve \( \displaystyle y'' + 4y' + 4y = 0 \) with \( \displaystyle y(0) = -3,\ y'(0) = 2 \).
Problem 6.159hard✓ Every equation proved
Solve \( \displaystyle y'' + 6y' + 9y = 0 \) with \( \displaystyle y(0) = 1,\ y'(0) = -1 \).
Problem 6.160hard✓ Every equation proved
Solve \( \displaystyle y'' + 2y' + 2y = 0 \) with \( \displaystyle y(0) = -1,\ y'(0) = -2 \).
Problem 6.161hard✓ Every equation proved
Solve \( \displaystyle y'' + 2y' + 1y = 0 \) with \( \displaystyle y(0) = -2,\ y'(0) = -2 \).
Problem 6.162hard✓ Every equation proved
Solve \( \displaystyle y'' + 6y' + 9y = 0 \) with \( \displaystyle y(0) = 2,\ y'(0) = -3 \).
Problem 6.179hard✓ Every equation proved
Solve \( \displaystyle y'' - 4y' + 8y = 0 \) with \( \displaystyle y(0) = -1,\ y'(0) = -2 \).
Problem 6.182hard✓ Every equation proved
Solve \( \displaystyle y'' + 2y' + 1y = 0 \) with \( \displaystyle y(0) = -2,\ y'(0) = 0 \).
Problem 6.183hard✓ Nihil obstat
Solve \( \displaystyle y'' - 6y' + 9y = 0 \) with \( \displaystyle y(0) = -1,\ y'(0) = 0 \).
Problem 6.203hard✓ Nihil obstat
Solve \( \displaystyle y'' + 2y' + 5y = 0 \) with \( \displaystyle y(0) = 1,\ y'(0) = 0 \).
Problem 6.204hard✓ Nihil obstat
Solve \( \displaystyle y'' - 4y' + 4y = 0 \) with \( \displaystyle y(0) = 1,\ y'(0) = -3 \).
Problem 6.205hard✓ Every equation proved
Solve \( \displaystyle y'' + 0y' + 4y = 0 \) with \( \displaystyle y(0) = 1,\ y'(0) = 2 \).
Problem 6.206hard✓ Every equation proved
Solve \( \displaystyle y'' - 1y' - 2y = 0 \) with \( \displaystyle y(0) = -3,\ y'(0) = -2 \).
Problem 6.207hard✓ Every equation proved
Solve \( \displaystyle y'' + 1y' - 2y = 0 \) with \( \displaystyle y(0) = -3,\ y'(0) = 0 \).
Problem 6.210hard✓ Every equation proved