Final answer:
To solve the IVP dy/dx = x^4(y-2), y(0)=5, we can use the method of separation of variables. The particular solution is ln|y-2| = (1/5)x^5 + ln|3|.
Step-by-step explanation:
To solve the initial value problem (IVP) dy/dx = x^4(y-2), y(0)=5, we can use the method of separation of variables.
We start by separating the variables, putting all terms involving y on one side and all terms involving x on the other side. This gives us (1/(y-2))dy = x^4dx.
Next, we integrate both sides of the equation. The integral of (1/(y-2))dy is ln|y-2|, and the integral of x^4dx is (1/5)x^5. So, we have ln|y-2| = (1/5)x^5 + C, where C is the constant of integration.
To find the particular solution, we substitute the initial condition y(0)=5 into the equation. This gives us ln|5-2| = (1/5)(0)^5 + C. Simplifying, we get ln|3| = 0 + C. Taking the exponential of both sides, we find |3| = e^C, which gives us the value of C. Therefore, the particular solution is ln|y-2| = (1/5)x^5 + ln|3|.