What is a solution to the differential equation #dy/dx=sqrt(xy)sinx#?

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Answer 1 · 2017-01-08T11:39:29

#y = (1/2int x^(1/2)sinxdx)^2#

We restrict ourselves to #x > 0#, #y > 0# so that:

#sqrt(xy) = sqrt(x) sqrt(y)# and the variables are separable:

#(dy)/(dx) = sqrt(xy)sinx = sqrt(x) sqrt(y) sinx#

#(dy)/sqrt(y) = sqrt(x)sinxdx#

#int y^(-1/2)dy = int x^(1/2)sinxdx#

#2sqrt(y) = int x^(1/2)sinxdx#

The right hand integral can be reduced to a Fresnel integral and cannot be expressed through elementary functions.