Answer
$1.5275$
Work Step by Step
In order to determine the exact solution, isolate the x and y terms on one side and integrate both sides.
$\int y dy=\int \sqrt x dx$ ...(1)
In order to determine the differential equation, we will have to take the derivative of the differential equation.
Equation (2) gives: $(\dfrac{y^2}{2})=(2/3)x^{3/2}+c$ ...(2)
Apply the initial conditions, to calculate the value of $c$
$\dfrac{(1)^2}{2}=(2/3)(0)^{3/2}+c\implies c=\dfrac{1}{2}$
Now, the particular solution is as follows:
$(\dfrac{y^2}{2})=(\dfrac{2}{3})x^{3/2}+\dfrac{1}{2}$ or, $y=\sqrt {\dfrac{4}{3}(x^{3/2})+1}$
Thus, the exact solution is: $y(1)=\sqrt {\dfrac{4}{3}((1)^{3/2})+1} \approx 1.5275$
