00:01
In this question, given that x plus 2 whole square, d .y of d x is equal to 5 minus 8y minus 4 xy and y 0 is equal to 1.
00:23
We have to find out initial value.
00:35
So now starting with the question, x plus 2 whole square, dy is equal to 5 minus 8y minus 4xy d x now x plus 2 hold square dyy minus 5 minus 8y minus 4 xy d x is equal to 0 so this can be written as minus 5 plus 8y plus 4xy d x plus x plus 2 hold square dy is equal to 0 and this is equation 1.
01:24
So comparing this equation with m -d -x plus n -d -y is equal to 0.
01:42
So, m is equal to minus 5 plus 8y plus 4xy and n is equal to x plus 2 whole square.
01:58
Now, del m of del y is equal to 8 plus 4x and del n of del x is equal to 2x plus 2.
02:15
Delm of del y is not equal to del n of del x, hence given differential equation is not exact.
02:44
Now 1 divided by n del m of del y minus del n of del n of del x is equal to 1 divided by x plus 2 whole square in bracket 8 plus 4x minus 2x minus 4.
03:13
Now this is equal to 1 of x plus 2 whole square and this value is 2x plus 2 4 2 is common so this number is x plus 2 and x plus 2 whole square so this is cancelled out so this value is equal to 2 divided by x plus 2 and this value is equal to fx now integrated factor is equal to e to the power integration of f x d x so this value is equal to 2 divided by x plus 2 d x so this is equal to x plus 2 whole square because e to the power ln f x is equal to f x now to make differential equation 1 to be exact multiplying equation 1 with integration factor, then we get minus 5 plus 8y plus 4xy, x plus 2 whole square, dx plus x plus 2 whole square, x plus 2 whole square, dy is equal to 0.
05:14
So, m is equal to minus 5 plus 8y plus 8y plus 4 xy, x plus 2 whole square and n is equal to x plus 2 whole power 4 so the general equation of exact differential equation is given by integration of m d x plus integration with n free of x d ,y is equal to c.
06:10
And here, y is equal to constant.
06:18
Now, this is equal to integration of y is equal to constant, minus 5 plus 8y plus 4xy, x plus 2 whole square, dx plus 0, dy is equal to c.
06:46
So whole this value is equal to 0...