Final answer:
To solve the given initial-value problem dy/dx = x^4y, y(0) = 7, separate variables and integrate both sides. Apply the initial condition to find the constant of integration and obtain the solution.
Step-by-step explanation:
To solve the initial-value problem dy/dx = x^4y, y(0) = 7, we can separate variables and integrate both sides. Rearranging the equation, we have: (1/y) dy = x^4 dx. Integrating both sides gives: ln|y| = (1/5)x^5 + C, where C is a constant of integration.
Next, we can use the initial condition y(0) = 7 to find the value of the constant C. Substituting x = 0 and y = 7 into the equation, we have: ln|7| = 0 + C, so C = ln|7|.
Therefore, the solution to the initial-value problem is: ln|y| = (1/5)x^5 + ln|7|. You can simplify this further if desired.
To solve the initial-value problem dy/dx = x^4y with the initial condition y(0) = 7, separate variables and integrate both sides. The resulting equation involves (y) and (x). Apply the initial condition (y(0) = 7) to determine the constant of integration. This process yields the specific solution to the differential equation, providing a mathematical expression for (y) in terms of (x) that satisfies both the given differential equation and the initial condition.