Final answer:
To solve the differential equation x(dy/dx) - 4y = 0, we can separate variables and integrate both sides. The general solution to the equation is y = Ax^4, where A is a constant.
Step-by-step explanation:
The given equation x(dy/dx) - 4y = 0 is a first-order linear differential equation. To solve this equation, we can separate the variables and integrate both sides. Rearranging the equation gives (dy/y) = (4dx/x). Integrating both sides gives ln|y| = 4ln|x| + C, where C is the constant of integration. To find the general solution, we can exponentiate both sides to get |y| = e^(4ln|x| + C). Using the properties of logarithms, this simplifies to |y| = e^C * e^(4ln|x|). We can further simplify this expression by writing e^C as a constant A, and using the property e^(4ln|x|) = (e^(ln|x|))^4 = x^4. Therefore, the general solution to the differential equation is y = Ax^4, where A is any constant.