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

1 Answer
Jan 8, 2017

y = (1/2int x^(1/2)sinxdx)^2y=(12x12sinxdx)2

Explanation:

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

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

(dy)/(dx) = sqrt(xy)sinx = sqrt(x) sqrt(y) sinxdydx=xysinx=xysinx

(dy)/sqrt(y) = sqrt(x)sinxdxdyy=xsinxdx

int y^(-1/2)dy = int x^(1/2)sinxdxy12dy=x12sinxdx

2sqrt(y) = int x^(1/2)sinxdx2y=x12sinxdx

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