Final answer:
The solution to the initial value problem x * dy/dx = y, y(1) = -1 is y(x) = -1/x.
Step-by-step explanation:
To find the solution to the initial value problem x * dy/dx = y, y(1) = -1, we can rewrite the equation as dy/y = dx/x. Integrate both sides to get ln|y| = ln|x| + C, where C is the constant of integration. Applying the initial condition y(1) = -1, we can solve for C and obtain the final solution y(x) = -1/x. This solution highlights the logarithmic relationship between x and y in the differential equation, emphasizing the role of initial conditions in determining the integration constant, crucial in deducing precise solutions for differential equations governing diverse phenomena in mathematics and scientific contexts.