Question

Find the equation of a curve passing through the point (0, 0) and whose differential equation is y' = e^x sin x.

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Solution

The given differential equation is
y' = e^x sin x or

dy over dx space equals space straight e to the power of straight x space sin space straight x

Separating the variables, we get,

dy space equals space straight e to the power of straight x space sinx space dx

Integrating,

integral space dy space equals space integral straight e to the power of straight x space sinx space dx

therefore space space space space space space space space space space space space space space space space space space straight y space equals space fraction numerator 1 over denominator 1 plus 1 end fraction straight e to the power of straight x space left parenthesis sinx minus space cosx right parenthesis space plus space straight c

open square brackets because space space integral straight e to the power of ax space sin space straight b space straight x space dx space equals space fraction numerator 1 over denominator straight a squared plus straight b squared end fraction straight e to the power of ax space left parenthesis straight a space sin space bx space minus space straight b space cos space bx right parenthesis close square brackets

therefore space space space space space straight y space space equals 1 half straight e to the power of straight x left parenthesis sinx space minus space cosx right parenthesis space plus straight c

...(1)
Now curve passes through (0, 0)

therefore space space space 0 space equals space 1 half straight e to the power of 0 left parenthesis sin space 0 space minus cos space 0 right parenthesis space space plus straight c space space space rightwards double arrow space space space 0 space equals space 1 half cross times space left parenthesis 0 minus 1 right parenthesis space plus straight c

therefore space space space space space space space space straight e space equals space 1 half

therefore space space space from space left parenthesis 1 right parenthesis comma space space straight y space equals space 1 half straight e to the power of straight x left parenthesis sinx space minus space cosx right parenthesis space plus space 1 half comma space which space is space required space solution. space

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