1.
Solve the system x' = 2x, y' = -3y. Give the general solution.
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Answer:
The two equations are not linked. Solve each one alone with a single exponential.
Linear systems practice problems
You solve two linked rates at the same time. Each rate uses both unknowns, for example x' = x + 2y and y' = 2x + y. The plan is fixed: find eigenvalues from a quadratic, find eigenvectors, then build the solution from exponentials, or from cosines and sines when the eigenvalues are complex. Recognition cue: two unknown functions, and each derivative equation mixes both of them. You will practice: Characteristic equation, Real eigenvalue solutions, Complex eigenvalue solutions, Phase portrait classification.
2.
Find the eigenvalues of the coefficient matrix of the system x' = 4x + y, y' = 2y.
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Answer:
The trace is 6 and the determinant is 8.
3.
Rewrite y'' + 5y' + 6y = 0 as a system of two first order equations. Use x1 = y and x2 = y'.
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Answer:
By definition x1' = y' = x2. The original equation gives y''.
4.
Find the eigenvalues for the system x' = 2x + y, y' = x + 2y.
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Answer:
The trace is 4 and the determinant is 4 - 1 = 3.
5.
The system x' = x + 2y, y' = 2x + y has eigenvalue λ = 3. Find an eigenvector v = (p, q) with p = 1.
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Answer:
Substitute λ = 3 into the first row equation.
Take p = 1, so v = (1, 1).
6.
Find the general solution of the system x' = x + 2y, y' = 2x + y.
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Answer:
The eigenvalues are 3 and -1. The eigenvectors are (1, 1) and (1, -1).
7.
Find the eigenvalues for the system x' = y, y' = -4x.
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Answer:
The trace is 0 and the determinant is 4.
8.
The system x' = x + 2y, y' = 2x + y has eigenvalues 3 and -1. Classify the origin as a node, a saddle, or a spiral.
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Answer:
The eigenvalues are real with opposite signs. That is a saddle.
9.
Classify the origin for the system x' = -4x + y, y' = 2x - 3y. State the type and the stability.
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Answer:
The trace is -7 and the determinant is 12 - 2 = 10.
Both eigenvalues are real and negative, so the origin is a stable node.
10.
Water flows from tank A to tank B. The salt amount x(t) in tank A satisfies x' = -2x. The salt amount y(t) in tank B satisfies y' = 2x - 3y. At t = 0, x = 3 and y = 0. Find x(t) and y(t).
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Answer:
The eigenvalues are -2 and -3. The eigenvectors are (1, 2) and (0, 1).
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More differential equations practice
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