Final answer:
To solve the differential equation dy/dx + (y/x) - sqrt(y) = 0 with the initial condition y(1) = 0, we use the method of exact differential equations. We find the function F(x, y) that satisfies the two partial differential equations and obtain the solution x - 2/3y^(3/2) + 1 = 0.
Step-by-step explanation:
The given differential equation is dy/dx + (y/x) - sqrt(y) = 0, with initial condition y(1) = 0. To solve this equation, we will use the method of exact differential equations. We rearrange the equation to isolate dy/dx and rewrite the equation as dx + x(dy/dx) - x^(3/2)sqrt(y) = 0. This equation can be written in the form M(x, y)dx + N(x, y)dy = 0, where M(x, y) = 1, N(x, y) = -xsqrt(y), and dM/dy = dN/dx = -x(sqrt(y))/2. Therefore, the given equation is exact.
To solve the exact equation, we need to find a function F(x, y) such that dF/dx = M(x, y) and dF/dy = N(x, y). Integrating dF/dx = M(x, y), we get F(x, y) = x + g(y), where g(y) is an arbitrary function of y. Now, differentiate F(x, y) with respect to y and equate it to N(x, y). We get dF/dy = g'(y) = -xsqrt(y). Integrating g'(y) = -xsqrt(y), we can find g(y). Integrating g'(y) = -xsqrt(y) with respect to y, we get g(y) = -2/3y^(3/2) + C, where C is the constant of integration. Therefore, F(x, y) = x - 2/3y^(3/2) + C.
Finally, by setting F(x, y) equal to a constant, we can find the solution to the original differential equation. In this case, using the initial condition y(1) = 0, we have x - 2/3(0)^(3/2) + C = x + C = 1, which gives C = 1. Therefore, the solution to the given differential equation is x - 2/3y^(3/2) + 1 = 0.