Section 1

1. Solve the given differential equation by separation of variables
dy-(y-1)² dx=0; y(0)=2

2. Solve the given differential equation by exact method. (1+lnx+y/x )dx-(1-lnx)dx=0
Hint ∫ln?x dx = xlnx-x

3. Solve the given differential equation by finding an appropriate integrating factor

y(x+y+1)dx+(x+2y)dy=0

4. Find the general solution to the given Bernoulli equation.

xy' + 4y = x⁴ y², y(2) = -1

5. a) Determine an integrating factor for the given problem

b) Solve the equation using linear technique y' = 2y + x² + 5

6. Find inverse Laplace transform of the given function. Show the details of your work and all calculations.

s/(s² + 4s + 5)

7. Find Laplace transform of the given function. Show the details of your work and all calculations.

8. Use Laplace transforms of integrals is find f(t) if L{F(s)} is. Show all details of your work

(3s+4)/(s⁴ + 16s² )

9. Showing all details of your work. Solve the following linear system by Gauss-Jordan Elimination technique.

(2x+y+2z = -1

x+z=-1

-x+3y-2z=7)

10. Showing all details of your work. Find the inverse of matrix A. make sure to check answer at the end.

11. Showing all details of your work. Find the inverse of matrix A. make sure to check answer at the end.

a) Graph the given: f(x) = x²,-1b) Determine whether the function is even, odd, or neither. Must show all details of your work
c) Find it Fourier series and the first four partial sums.

Section 2

Show All Working to get full Credit

1. Solve the boundary-value problem using separation of variables technique.

dy/dx = (3x² + 4x - 4)/(2y - 4), y (1)=3

2. Solve the boundary-value problem using separation of variables technique.

ln(x) dy/dx= xy, , y (1)=2

3. a) Determine the integrating factor needed to make the given equation exact.

b) Solve the equation using exact technique.

6xy dx+(9x²+4y)dy=0

4. Solve the equation using linear technique.
a) (x+1) y¹ + y = lnx, y(1)=10
b) xy¹ - y = 4xln(x), y(0,1)=8

5. Find the general Solution to the given Bemoulli Equation
a)t².dy/dt + y² = ty
b) y'- 4/x.y = x√(y,) y(1) = 4

6.

a) Derive the differential equation representing the change in the given circuit. must show all required steps to earn credit.
b) Determine the damping of circuit.
c) Solve for change in circuit. Assume zero initial charge
Solve above questions for two Pictures below

1.

2.

Section 3

Show all details of your work for credit

1. Use Laplace transform of derivatives to find the transform of
f(t) if f(t)=  cos²2t

2. Find the transform of the given function

3. Solve the shifted data IVP by Laplace transform
 a) y'' - 2y' - 3y = 0, y(4) = 3, y' (4) = -17
 b) y'' - y' - 6y = 0, y(-1) = 4, y' (-1) = 12

4. Solve the IVP using the Laplace transform
 a) y'' - y' = e^t cos t, y(0)=0, y' (0)=0
 b) y'' - 3y = 13 sin2t, y(0)=6

5. Use Laplace transform to find the initial value problem for current in the given RLC-circuit assume zero initial current and charge.

a)

b)

6. A) Show all details of your work, determine the following if A, B and C are: solve both TOP and Bottom
i. c^T Ac
ii.[(A+A^T)(B-B^T)]c

B ) Show all details of your work, determine the following if A, B and C are

i. c^TAc
ii. |(AB)²|
iii. (A+A^T)(B-B^T)

iv

7. Showing all details of your work. Solve the following linear system by Gauss Elimination technique.

(2x₁ + x₂ + 2x₃ = -1

x₁ + x₃ = -1

-1x₁ + 3x₂ - 2x₃ = 7)

Section 4
a) Showing all details of your work. Find the inverse of matrix A by Gauss-Jordan Elimination.

b) Showing all details of your work. Find the inverse of matrix A by Gauss-Jordan Elimination

a) Showing all details of your work. Find the inverse of matrix A by determinates and Cofactors.

b) Showing all details of your work. Find the inverse of matrix B by determinates and Cofactors.

3 a) Showing all details of your work. Use Cramer's Rule and Determinants to find the

currents in the given system of equations.

(5i₁ + 10i₃ = 220

20i₂ + 10i₃ = 240

i₃ = i₁ + i₂ )

4. b) Showing all details of your work. Use Cramer's Rule and Determinants to find the currents in the given system of equations.

(x + y -2z = 14

2x - y + z = 0

6x + 3y + 4z = 1)

5. a) Find the eigenvalues and eigenvectors of matrix A. make sure check answer at the end

6. b) Find the eigenvalues and eigenvectors of matrix A. make sure check answer at the end

7. Showing all details of your work, determine the following if A,B, and c are:

a) |(AB)² |

b) (A+B^T)(c)

8. a) Graph the given function :

f(x) = { (1, -5

            1+x, 0≤x<5)

b) Determine whether the function is even, odd, or neither. Must show all working

c) Find its Fourier series and the first four partial sums