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Math 287 Chapter 1 Exam
- For each of the following differential equations: Find the order of the equation, Identify the equation as separable or not, Identify the equation as linear or not, If the equation is linear then identify if the equation is homogeneous or nonhomogeneous Equation Order Separable or Not
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
- (a) Use Euler’s Method with a step size h = 2 to approximate , where is a solution to the IVP
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?
- y (^) 1 ( t )= et and y (^) 2 ( t )= e^2^ t are both solutions to y ′ ′^ − 3 y ′+ 2 y = 0. Further, y (^) p ( t )= t + 2 is a solution to the equation y ′′ − 3 y ′+ 2 y = 2 t + 1 (a) Write the general solution to y ′′^ − 3 y ′+ 2 y = 2 t + 1 (b) Find the solution to the IVP y ′′^ − 3 y ′+ 2 y = 2 t + 1 , y ( 0 )= 6 , y ′( 0 )=− 2
- All parts of this problem refer the differential equation y ′^ =( y + 1 )( y − 2 ) (a) Find the equilibrium solutions to (b) Sketch the direction field of this this differential equation differential equation. Clearly indicate the equilibrium solutions on this direction field (c) Graph the solution to the IVP y ( 0 )= 1 in the direction field you drew in part (b) (d) Solve the differential equation, finding all nonequilibrium solutions. Write your answer in explicit form (as apposed to implicit form) to receive full credit.
- (a) Find the explicit form of the general (b) Find the solutions to the initial value solution to y ′^ = te^2 t − y problem y ′^ = te^2 t − y , y ( 0 )=− 1
- (a) Find the explicit form of the general (b) Find the solutions to the initial value solution to t^2 y ′ + ty = t^3 problem t^2 y ′ + ty = t^3 , y ( 1 )= 4
- A 50-gallon tank initially contains 10 gallons of fresh water. At time , a brine solution containing 1 lb of salt per gallon is poured into the tank at the rate of 4 gallons per minute while the well stirred mixture leaves the tank at the rate of 2 gallons per minute. Let t be the number of minutes after the brine solution begins to enter the tank up until the tank overflows
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
- (EXTRA CREDIT)
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.
- For each differential equation find a matching solution graph and matching phase plane trajectory x ^ + 9 x = 0 x ^ + 9 x =sin( 3 t ) x ^ + 2 x + x = 0 x ^ + 2 x + x = t xt Solution Graph xx trajectory
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 ( )
(e) The span of a set { 2, 0, − 1 , −3, 1, 4}
8 pts 3. and
A
= ⎢^ − ⎥
1
A −
. 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^32
B^14
= ⎡^ −
C^0
(a) A + B (b) AC (c) 5 A (d) CT
8 pts 5. ⎥
K
K
A
(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 )
(b) T : P 2 → R^2 by T ( p ( t ))= p ( 0 ), p ( 1 ). Use S = { 1 , t , t^2 }as the standard basis for P 2
10 pts 10. Let T : V → W be a linear transformation. Prove that if { T v ( 1 ), T v ( 2 ), T v ( 3 )}is a linearly
independent, then { v 1^ ,^ v 2^ , v 3^ }is a linearly independent set. For each step of your prove give your reason
why the step is valid. Exam 4
- Match the phase planes to the locations in the parameter plane ( Note: one phase plane will have NO match and one location will have NO match) ( 8 points ) A
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
