MATH 2410 Exam I Fall 2022
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
MATH 2410 Exam I
Fall 2022
1. [15 pts.] Consider the diferential equation 
(a) (4 pts) Find the equilibrium points of this diferential equation.
(b) (4 pts) Sketch the graph z = y(y2 - 4). Use this information to draw the phase line (with arrow direction indicating how solution changes as x increases).
(c) (7 pts) Sketch the graphs of the solutions that satisfy the initial conditions
(i) y(0) = 3; (ii) y(0) = 1; (iii) y(0) = -2. (Clearly label which is which.)
The horizontal axis is x while the vertical axis is y.
2. [20 pts.] Consider the differential equation 
(a) (2 pts) Show that the equation is separable.
(b) (8 pts) Find the general solution to the equation.
(c) (2 pts) Find a solution that satisies y(1) = 2:
(d) (4 pts) Consider another initial condition y(0) = 0, can you ind a solution to the new initial value problem using part (a)? Explain what goes wrong in this case for the procedures in part (a).
(e) (4 pts) Is there a solution to the equation with the initial value y(0) = 0? What is the solution if it does? Explain.
3. [16 pts.]
(a) (13 pts) Find the general solution to the diferential equation 
(b) (3 pts) Find the solution if the initial condition y(0) = 1 is imposed.
4. [16 pts.] Consider the initial value problem y、= y + x with y(0) = 1. Use the Euler’s method to obtain the approximation of y( 3/2 ) with a step size h = 1/2.
5. [25 pts.]
(a) (5 pts) Find the general solution of y、、+ 4y、- 5y = 0.
(b) (5 pts) Find the general solution of y、、- 4y、+ 13y = 0.
(c) (9 pts) Solve y、、+ 4y、+ 4y = 0 with initial conditions y(0) = 1 and y、(0) = 0.
(d) (6 pts) Suppose the auxiliary equation of a linear homogeneous constant coe伍cient 4th order diferential equation Ly = 0 has the roots {1, 1, 4 + 2i, 4 - 2ig. What is the general solution to this diferential equation.
6. [8 pts.] Given the functions y1 = e2x and y2 = xe2x. Evaluate the Wronskian W (y1 , y2 ) at x = 1.
2023-10-19