00:01
Hello, i have a question.
00:01
And this i have to find the general solution of the following differential equation, d .y by dx equal to minus sine x into y plus 1 by 2, sine 2x.
00:24
So let us make it in the form of general differential, that is linear differential equation, d .y by dx plus px into y equal to qx.
00:35
So if we try to write it, it becomes dy by dx plus, if you take this left side, it will be plus sine x into y equal to 1 by 2, sine 2x.
00:52
This is in this form where i will assume p as a function of x equal to sine x after comparing and 1 by 2 s and k and q as a function of x equal to 1 by 2 sine 2 x okay let us get started and write integrating factor integrating factor be e days to the power integration of bx d x so e d x so e days to the power sine x d x so e raise to the power minus cos x will be the integrating factor now solution is y into integrating factor equal to qx into if into dx plus c where c is the constant of integration.
01:53
So y into e raise to the power minus cos x equal to qx is 1 by 2 sine 2 x into e raise to the power minus cause x dx plus c.
02:20
So if we could write it as 1 by 2, 2 sine x into cos x erase to the power minus cos x dx plus c, 2 and 2 would get cancelled out.
02:42
That means this is sine x, cos x, erase to the power minus cos x dx plus c.
02:50
Now let us assume e raise to the power minus cos x equal to t.
03:01
So it is to the power minus cos x and differentiation of minus cos x is sine x dx will be d t.
03:10
So this whole thing including this will be equal to d t...