To solve this equation using the method of variables separables, we need to rearrange it so that all the terms involving y are on one side and all the terms involving x are on the other side. We can do this by multiplying both sides by xy and then adding x^2 to both sides. This gives us:
Now we can divide both sides by (y2) and separate the variables. This gives us:
The next step is to integrate both sides with respect to their respective variables. This will give us two functions of one variable each, plus a constant of integration. We can use partial fractions to simplify the integrals. The left-hand side becomes:
The right-hand side becomes:
where C is an arbitrary constant. This is the general solution of the differential equation. To find a particular solution, we need to know an initial condition, that is, a pair of values (x0,y0) that satisfy the equation. For example, if we are given that y(1)=1, then we can plug in x=1 and y=1 into the general solution and solve for C. This gives us:
Therefore, the particular solution that satisfies y(1)=1 is:
This is the final answer. We can check that it satisfies the original differential equation by differentiating it with respect to x and simplifying. We can also plot the solution curve using a graphing calculator or software.
We hope this article has helped you understand how to solve ecuaciones 23 with Yu Takeuchis method. If you want to learn more about differential equations and other topics in mathematics, you can check out our other articles or visit our website for more resources. 04f6b60f66