DE x y dx +( x^2 + y ^2 ) dy =0
The given equation is a first-order homogeneous differential equation and can be solved using the method of separating variables.
To solve this equation, we need to separate the variables and integrate. Here's how you can do it step by step:
Step 1: Move the terms with 'dx' and 'dy' to opposite sides of the equation, and put all terms involving 'y' on one side:
(x^2 + y^2) dy = -dx
Step 2: Divide both sides of the equation by (x^2 + y^2):
dy/dx = -dx/(x^2 + y^2)
Step 3: Rewrite dy/dx as (1/y'):
1/y' = -dx/(x^2 + y^2)
Step 4: Multiply both sides by y' to get rid of the denominator:
y' = -y^2 dx/(x^2 + y^2)
Step 5: Rearrange the equation:
y'/(y^2) = -dx/(x^2 + y^2)
Step 6: Integrate both sides of the equation:
∫(1/y^2) dy = -∫(dx/(x^2 + y^2))
The integral on the left side can be evaluated as:
-1/y = -arctan(x/y) + C1
Here, C1 is the constant of integration.
Step 7: Multiply both sides by -1 to solve for y:
y = -1/(-arctan(x/y) + C1)
Simplifying further,
y = 1/(arctan(x/y) - C1)
So, the solution to the given differential equation is y = 1/(arctan(x/y) - C1), where C1 is the constant of integration.
Note: The integration in Step 6 requires advanced calculus techniques, so the exact integral may not always have a simple closed-form expression. In such cases, the solution can be expressed in terms of implicit equations.