Solution we have m x y 2 xy 3 x 2 and n x y x 2 2 y

This preview shows page 6 out of 6 pages.

Solution: We have M ( x, y ) = 2 xy - 3 x 2 and N ( x, y ) = x 2 - 2 y , then ∂M ∂y = 2 x = ∂N ∂x . Thus the equation is exact. (b) Find the solution to (**) satisfying the initial condition y (1) = 1. Solution: Since the equation is exact we know that all the solutions are of the form Ψ( x, y ) = c, where Ψ( x, y ) is a function satisfying Ψ ∂x = M, Ψ ∂y = N. Thus Ψ( x, y ) = Z M ( x, y ) dx = Z 2 xy - 3 x 2 dx = x 2 y - x 3 + C ( y ) , where C ( y ) is a function depending only on the variable y . To find C ( y ) we take the partial derivative of Ψ( x, y ) with respect to y and compare with N ( x, y ). We obtain Ψ ∂y = x 2 + C 0 ( y ) N ( x, y ) = x 2 + C 0 ( y ) x 2 - 2 y = x 2 + C 0 ( y ) - 2 y = C 0 ( y ) - y 2 + c = C ( y ) . Therefore the solutions to the exact equation are of the form x 2 y - x 3 - y 2 = c. Since y (1) = 1 we must have (1) 2 (1) - (1) 3 - (1) 2 = c and so the solution to the IVP is given by x 2 y - x 3 - y 2 = - 1 . Page 6
Image of page 6

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern