Question

A) In this problem we consider an equation in differential form ???+???=0Mdx+Ndy=0. The equation

(5?^4?+4cos(2?)?^−4)??+(5?^5−3?^−4)??=0

in differential form ?˜??+?˜??=0 is not exact. Indeed, we have

My-Mx= ________ 5x^4-4(4)cos(2x)*y^(-5)-25x^4

(My-Mn)/M=_____ -4/y

in function y alone.

?(?)= _____y^4

Multiplying the original equation by the integrating factor we obtain a new equation ???+???=0 where

M=____

N=_____

which is exact since

My=_____

Nx=______

This problem is exact. Therefore an implicit general solution can be written in the form  ?(?,?)=? where

?(?,?)=_________

Finally find the value of the constant ?C so that the initial condition ?(0)=1y(0)=1 is satisfied.

C=____________

found first 3 blanks need help with rest

B)

In this problem we consider an equation in differential form ???+???=0 The equation

(4?^−3?−(12?^2?^3?^−?+2?^−?sin(?)))??+(−(12?^3?^2?^−?+12?^−3?))??=0

1n differential form ?˜??+?˜??=0 is not exact. Indeed, we have

?˜?−?˜?=______ -(12x^3y^2e^-x+12e^(-3y))

(My-Nx)/N=____1

can be considered as a function of  x alone.

Namely we have ?(?)=____e^x

Multiplying the original equation by the integrating factor we obtain a new equation ???+???=0 where
M=4e^(-3y)e^(x)-(12x^2y^3+2sin(x))

N=__

which is exact since

My=___

Nx=___

are equa;

This problem is exact. Therefore an implicit general solution can be written in the form  ?(?,?)=?where

F(x,y)= _____

Answer 1