Question

solve the initial value problem dy/dx = x⁶(y-5), y(0) = 7 , question content area bottom part 1 the solution is enter your response here. (type an implicit solution.

Answer 1

To solve the given initial value problem dy/dx = x⁶(y-5), y(0) = 7, we can separate variables and integrate both sides. The implicit solution is y = 5 + A e^[(1/7)x^7] or y = 5 - A e^[(1/7)x^7].

Step-by-step explanation:

To solve the initial value problem dy/dx = x^6(y-5) with the initial condition y(0) = 7, we can separate variables and integrate both sides.

First, we isolate the y terms on one side and the x terms on the other side:

dy/(y-5) = x^6 dx

Next, we integrate both sides:

ln|y-5| = (1/7)x^7 + C

Finally, we solve for y by eliminating the natural logarithm:

|y-5| = e^[(1/7)x^7 + C]

y - 5 = A e^[(1/7)x^7] or y - 5 = -A e^[(1/7)x^7] (where A = e^C)

So, the implicit solution to the initial value problem is either y = 5 + A e^[(1/7)x^7] or y = 5 - A e^[(1/7)x^7].