To solve the initial value problem given by the differential equation:
x² dy/dx = y + xy + 1
we can start by rearranging and simplifying the terms in the equation.
Step 1: Rearranging the Equation
First, we rewrite the equation for clarity:
dy/dx = (y + xy + 1)/x²
This equation can be addressed by separating variables. Reorganize it:
dy/(y + xy + 1) = dx/x²
Step 2: Integrating Both Sides
Now we can integrate both sides. We need to perform the integration on the left and right sides separately:
∫ dy/(y + xy + 1) = ∫ dx/x²
The integral on the right-hand side is straightforward:
∫ dx/x² = -1/x + C
Step 3: Solving the Left Integral
For the left-hand side, we can use substitution or partial fractions depending on the form of y + xy + 1. If we can express it in a simpler rational form, we can find an easier way to integrate it. Let’s say we can express it:
The actual form would be dependent on our terms, but if we find a simplifiable expression, we can thus find:
Step 4: Explicitly Solving for y
After integration, we would ideally end up with an expression for y in terms of x and C. This would involve some algebraic manipulation:
After applying the integration and back-substituting for C based on the initial condition given (like y(4)), we can derive the explicit form.
Step 5: Applying Initial Conditions
Using the condition, we substitute x=4 and y(y(4)) into our integral results to solve for C.
Final Result
Finally, we can combine our results, simplifying where necessary, to find the explicit solution for y based on the given initial condition.
Conclusion
This process guides how we would typically handle such an initial value problem. Each step, from rearranging and integrating to applying initial conditions, is essential to deriving a well-defined explicit solution.