if the differential equation of a curve, passing through `(0,-(pi)/(4))` and `(t,0)` is `cosy((dy)/(dx)+e^(-x))+siny(e^(-x)-(dy)/(dx))=e^(e^(-x))` then find the value of `t.e^(e^(-1))`

Answer

1

Explanation

We have `cosy((dy)/(dx)+e^(-x))+siny(e^(-x)-(dy)/(dx))=e^(e^(-x))`
`rArr (cosy-siny)(dy)/(dx)+(cosy+siny)e^(-x)=e^(e^(-x))`……..(1)
Let `cosy+siny=u`
`therefore` Equation (1) reduces to
`rArr (du)/(dx)+e^(-x)u=e^(e^(-x))`,
Which is linear differential equation whose solution is
`ue^(-e^-(x))=x`or `(cosy+siny)e^(-e^(-x))=x`
Putting (t,0), (putting (t,0))
`rArr e^(-e^(-t))=t`
`rArr t.e^(e^(-t))=1`