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Second-order linear equations practice problems

Constant coefficients: the characteristic equation and its three cases. 30 problems with worked solutions; in 30 of them every equation is proved by a computer algebra system.

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 \( \displaystyle y'' + 6y' + 9y = 0 \) 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 \( \displaystyle y'' + 2y' + 10y = 0 \) with \( \displaystyle y(0) = 1,\ y'(0) = 3 \).
Problem 6.37hard✓ 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