Solve each initial value problem, then classify the type
and stability of the critical point at (0,0) of each
system:
x1'=4x1-5x2; x2'=5x1-4x2
x1(2019)=16; x2(2019)=25
Classify (if possible) each critical point of the given plane
autonomous system as a stable node, a stable spiral point, an
unstable spiral point, an unstable node, or a saddle point. (Order
your answers from smallest to largest x, then from smallest to
largest y.)
x' = x(1 − x2 − 5y2)
y' = y(5 − x2 − 5y2)
10. Solve the following initial value problem:
y''' − 2y '' + y ' = 2e ^x − 4e^ −x
y(0) = 3, y' (0) = 1, y''(0) = 6
BOTH LINES ARE PART OF A SYSTEM OF EQUATIONS
(3 pts) Solve the initial value problem
25y′′−20y′+4y=0, y(5)=0, y′(5)=−e2.
(3 pts) Solve the initial value problem
y′′ − 2√2y′ + 2y = 0, y(√2) = e2, y′(√2) = 2√2e2.
Consider the second order linear equation t2y′′+2ty′−2y=0,
t>0.
(a) (1 pt) Show that y1(t) = t−2 is a solution.
(b) (3 pt) Use the variation of parameters method to obtain a
second solution and a general solution.
Consider the initial value problem
y′ = 18x − 3y, y(0) = 2
(a) Solve it as a linear 1st order ODE with the method of the
integrating factor.
(b) Solve it using a substitution method.
(c) Solve it using the Laplace transform.
Solve the given initial-value problem by finding, as in Example
4 of Section 2.4, an appropriate integrating factor.
(x2 + y2 ? 3)
dx = (y + xy)
dy, y(0) = 1