Final answer:
The differential equation in question is solved by separating variables, integrating both sides, and then applying the initial condition to find the specific integration constant. The particular solution is obtained with this constant.
Step-by-step explanation:
The student is asking for the particular solution to the differential equation \( \frac{dy}{dx}=\frac{x-4}{2y} \) given the initial condition \( y(6)=4 \). To find the particular solution, we first solve the separable differential equation by separating the variables.
We integrate both sides of the separated equation. To incorporate the initial condition, we substitute \(x = 6\) and \(y = 4\) into the general solution to find the value of the integration constant. Finally, we write the particular solution that satisfies the given initial condition.
Step-by-Step Solution:
- Multiply both sides by \(2y\) to separate the variables: \(2y dy = (x-4) dx\).
- Integrate both sides: \(\int 2y dy = \int (x-4) dx\).
- Find the constants of integration and use the initial condition \(y(6)=4\) to solve for the particular constant.
- Write the particular solution with the determined constant.