00:01
Hello, let's look at the question.
00:02
The given differential equation is xy dx minus x square plus 2y square dy is equal to zero.
00:12
So, the differential equation is in the form of m x dx plus n dy is equal to zero where m is equal to y and n is equal to minus x square minus 2y square.
00:27
So, a partial derivative for m is x and partial derivative for n is minus 2x.
00:38
So, clearly a partial derivative of n with respect to y is not equal to partial derivative of n with respect to x.
00:47
The given differential equation is not exact.
00:50
Now, integrating the factor 1 by mx plus ny, our integrating factor will be 1 by xy x plus y minus x square minus 2y square.
01:06
So, our integrating factor will be 1 by x square y minus x square y plus 2y cube which would be minus 1 by 2y cube and this will get cancelled out.
01:19
So, our equation multiplied by itself, integrating factor will be minus 1 by 2y cube multiplied by xy dx minus x square plus 2y square dy equals to zero.
01:36
It will be minus x by 2y square dx plus x square plus 2y square by 2y cube dy is equal to zero.
01:49
This is our equation 2.
01:53
Now, decreasing the form of m1 dx plus m1 dy equals to zero, our m1 will be minus x by 2y square and n is x square plus y square by 2y cube that is x square by 2y cube plus y by 2y, sorry 1 by 2y which is our equation 3.
02:24
So, our partial derivative of m with respect to y is minus x by 2 minus 2y by cube that is xy by cube and partial derivative of n1 with respect to x is 2x plus zero by 2y cube that will be x by y cube...