Here's how you can solve the given ordinary differential equation (ODE):
1. The given differential equation is:
(dy/dx) = x/(5 + x²)
This is an example of a first order ODE, specifically a variable separable ODE because we can rewrite it in a way that all terms involving x are on one side of the equation and all terms involving y are on the other.
2. Separate the variables:
We start by moving dx to the other side and isolating dy on one side:
dy = (x/(5 + x²)) dx
3. We now integrate both sides of the equation:
∫ dy = ∫ (x/(5 + x²)) dx
The left side is simple, it just integrates to y. The right side though may seem a bit complex, but by recognizing that the denominator is the derivative of x² + 5, we can perform this integration quite simply.
4. Evaluate the integral:
y = ∫ (x/(5 + x²)) dx
With a simple substitution of u = x² + 5, the integral on the right becomes straightforward and evaluates to:
y = 1/2 ln|x² + 5|
Adding the constant of integration C gives us our general solution.
5. Write the General solution:
Therefore, the solution of the given differential equation is y(x) = C + 1/2 log(x² + 5)
Here, C is a Constant, log denotes the Natural Logarithm, and y(x) is the function of x.
This general solution represents an infinite number of possible solutions, each corresponding to a different value of C. The specific solution will depend on the initial condition provided or other boundary values.
Remember that in each of these steps, we have applied standard rules of calculus. The steps involved rearranging the equation, separation of variables, and then integrating both sides. The integral needed us to spot the opportunity to use a simple substitute to make the integration more straightforward.