00:01
Hi, in this question in the a part, consider y -cube minus y -square into sine x minus x into dx plus 3 into y -square into x plus 2y into cos x into d -y.
00:21
This is equal to 0.
00:23
If we remove minus common from all this term, we can say that this can be written as x minus -y -cube.
00:31
Plus y square into sine x d x minus this will be written as 3x y square plus 2y into cos x d y that is equal to 0.
00:49
So that means we can say that your m b equal to x minus y cube plus y square into sine x and and n b equal to minus of 3x y square plus 2y into cos x and when we differentiate or we can say that when we take the partial derivative we get this as do m by do y is equal to do n by do x that is equal to minus 3y square plus 2y into sine x and hence then this is an exact solution or exact equation.
01:46
So therefore we have g of x comma y to be equal to minus integral of 3xy square plus 2y into cos x, dy, y.
01:59
G x is equal to minus y cube plus y square into to sine x plus h dash of x this is equal to x minus y cube plus y square into sine x that is equal to m.
02:19
Here h dash of x is equal to x so h of x is equal to x square by two.
02:27
Therefore we get g of x comma y to be equal to minus x y cube minus y squared into cos x plus this will be equal to x squared divided by 2.
02:42
Therefore, the general solution is minus x into y cube minus y square into cos x plus x squared by 2 is equal to g of x comma y.
02:56
In the b part we have been given, y dash plus y is equal to 1 minus e raise to minus 2x divided by e -raise 2x plus e -raise 2 minus x comparing with y -dash plus p of x into y is equal to q of x into y raise to n here p of x is equal to 1 and q of x is equal to 1 minus e raise to minus 2x divided by e raised to x plus e raised to minus x...