To solve the given differential equation, we must first recognize that it's in a form of a first-order differential equation, which can be written as dy/dx = f(x, y).
Here, our function f(x, y) = x√y. This is considered as a separable differential equation because we can rearrange terms in order to use simple integration to solve it. Here's the step-by-step process:
1. Separate the variables by bringing the relevant terms together and rewrite √y as y^(1/2):
x dx = y^(-1/2) dy
2. Now, we integrate both sides with respect to their variables.
∫x dx = ∫y^(-1/2) dy
3. After the integration, the equation should look like this:
x^2/2 = 2y^(1/2) + C1, where C1 is the constant of integration. Notice that we use a new constant C1 because it may be different from the constant C we would get if we were to integrate the left side.
4. Now, clearance of fraction by multiplying both sides by 2 yields
x^2 = 4*y^(1/2) + 2C1
5. Raise both sides of the equation to power of 2 to remove the square root.
(x^2)^2 = (4*y^(1/2) + 2C1)^2
6. Simplify both sides to get the equation in terms of y = f(x):
y(x) = (C1^2)/4 + (C1*x^2)/4 + x^4/16
This represents the general solution to the given differential equation.
Ans : y(x) = (C1^2)/4 + (C1*x^2)/4 + x^4/16