To solve the initial value problem given by the differential equation:
xy² dx + 2xy x² + 5 dy = 0
with the initial condition y(1) = 1, we start by rearranging the equation:
1. First, rewrite the equation in a more standard form:
dy/dx = – (xy²)/(2xy x² + 5)
2. To separate variables, express it as:
(2xy x² + 5) dy + xy² dx = 0
3. Now we separate variables:
dy = – (xy²)/(2xy x² + 5) dx
4. Integrate both sides accordingly:
∫ dy = – ∫ (xy²)/(2xy x² + 5) dx
5. Perform integration on the left side:
y = – ∫ (xy²)/(2xy x² + 5) dx + C
6. After integrating and simplifying (the integration may require some techniques like substitution, depending on the complexity), apply the initial condition y(1) = 1 to find the constant C.
7. Substitute x = 1 and y = 1 into the equation to solve for C.
8. Finally, express the general solution using the computed constant. Each step may involve algebraic manipulation and integration techniques suited to the form of the differential equation you derived.
This process will allow you to solve the initial value problem step by step.