Answer each of the following to the best of your ability. Show all necessary wor

Question

Answer each of the following to the best of your ability. Show all necessary work and follow all instructions carefully. Use this sheet as a cover sheet and attach your work to the back of this sheet. Remember, I am testing you, not your calculator. Each question is worth 2 points. Good luck! Use separation of variables to solve the initial value problem: dy/dx = csc^2y/x^3, y(pi) = 0. show that the following differential equation is exact and then solve the equation: [2xsin(x^2) + 6xe^5y]dx + (15x^2e^5y + 15y^2) dy = 0. Use a linear substitution to solve the differential equation: dy/dx = (5x + y)^3 -5, y(1) =0, express your answer in "x = " form. Solve the linear differential equation (x+3dy/dx = 5x^2 - 2y Using an integrating factor. You may assume that x >0. Determine where dy/dx=x/y + x-sin(x/y)will have unique solutions.

Explanation / Answer

1) given

dy/dx =csc2y /x3

seperate the variables

dy/csc2y =x3 dx

sin2y dy=x3 dx

(1/2)(1-cos2y) dy=x3 dx

integrate on both sides

(1/2)(1-cos2y) dy=x3 dx

(1/2)(y-(1/2)sin2y)=(1/4)x4 +c

y-(1/2)sin2y=(1/2)x4 +c

given y()=0

0 -0 =(1/2)4 +c

c=-(1/2)4

y-(1/2)sin2y=(1/2)x4 -(1/2)4 is the particular solution

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