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A set of differential equations with instructions to find their orders, separability, linearity, and homogeneity. It includes specific equations with given initial conditions and asks to approximate solutions using euler's method and the fourth-order runge-kutta method.
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Math 287 Chapter 1 Exam
Linear or Not Homogeneous or Not yy ′^ = x ( y^2 + 1 ) Q t dt t
dQ (^) 3 sin 10 2
y ′′^ − 3 y ′+ 2 y = 0
y ( 5 ) y ( t ) y ′ = 5 yt + y^2 y ( 1 )= 2 (b) Suppose the error in the approximation from part (a) is 6.0. How large would you expect the error in the approximation to be if the step size were changed to h =0.5? (c) Suppose using Fourth-Order Runge-Kutta with a step size of h = 2 to approximate , where is a solution to the IVP
y ( 5 ) y ( t ) y ′= 5 yt + y^2 y ( 1 )= 2 produces an error of 1.0. How large would you expect the error in the approximation to be using Fourth-Order Runge-Kutta if the step size were changed to h =0.5?
t = 0
(a) Set up an initial value problem that models (b) Solve the IVP in part (a) to find the the gallons of solution in the tank at time t gallons of solution in the tank at time t (c) Set up an initial value problem that models the pounds of salt in the tank at time t (d) Solve the IVP in part (c) to find the pounds of salt in the tank at time t (e) Find the amount of salt in the tank at the moment the tank overflows
2 1 2
− 3 y = ty − dt t
The equation dy is an Euler-Homogeneous equation and a Bernoulli equation.
(a) Using the change of variables y = vt , where is an unknown function of , transform the differential equation into a separable equation and separate the variables.
v t
(b) Using the change of variables v = y^2 transform the equation into a linear equation of the form p ( t ) v q ( t ) dt
dv (^) + =
Exam 2 4.1 – 4.6, 8.1, - 8.
xt Solution Graph (I) (II) (III) (IV)
xx Trajectory (I) (II) (III) (IV)
10 pts 2. Identify each of the following as a vector space or a coset. Next, if the set is a vector space determine its dimension. If the set is a coset determine the dimension of the associated subspace.
(a) The line through f ( ) t = t − 1 in the direction g t ( ) = et (b) The solution set to y ′′′^ − 2 y ′′^ + y ′− y = 0 (c) The solution set to differential equation x + 2 x + x =cos( ) t (d) The kernel of T : C^1 [0,1] → C [0,1] by T ( f ) = f ′′( ) t − f t ( )
8 pts 3. and
1
. Use this information to solve the two following systems.
x y z (a) x y z 1 x y z
(b)
x y z x y z x y z
4. For matrices (^) ⎥ and calculate each of the following: ⎦
(a) A + B (b) AC (c) 5 A (d) CT
8 pts 5. ⎥
(a) Calculate the determinant of A (b) Determine the values of K for which matrix A is singular
15 pts 6. Use Gauss-Jordan Elimination to find the RREF of the augmented matrix for the system and the solution set to the system of equations.
x y z
x y z
x y z
4 pts 7.(EXTRA CREDIT) The solution set to problem 6 is a coset of some subspace of R^3. Find a basis of this subspace.
22 pts 8. T : R^4 → R^4 by T ( x G^ )= A ⋅ x G. Matrix A and the row echelon form of are shown below
A row echelon form of A ⎥
(a) Find a basis for the range of T (b) Find the dimension of the range of T (c) Find a basis for the kernel of T (d) Find the dimension of the kernel of T (e) Is T a projection of R^4 onto the range of T. Show that your answer is correct.
8 pts 9. Find the standard matrix for each transformation (a) T ( x , y , z )=( x + 2 y , x − 2 y , x + y − 2 z , 2 y + 6 z )
why the step is valid. Exam 4
Tr A
25 pts 6. The matrix (^) ⎥has eigenvalues ⎦
− 3 and 1 with corresponding eigenvectors and
dx dt dy dt
x y t x y
Use this information to solve the system of equations