Question

Solve the differential equation (no initial condition): dy/dx = x⁸ / 1+y⁶ Please provide a step-by-step solution for full credit.

Answer 1

To solve the differential equation dy/dx = x⁰ / (1+y⁶), separate the variables and integrate both sides to find the general solution, y = 7x + 7C.

Step-by-step explanation:

To solve the differential equation dy/dx = x⁰ / (1+y⁶), we need to use separation of variables. This involves rearranging the equation so that all the y terms are on one side and all the x terms are on the other side. Here are the steps:

1. Separate the variables by multiplying both sides by (1+y⁶) and then dividing by dy:
2. (1+y⁶) dx = x⁰ dy
3. Integrate both sides of the equation:
4. ∫ (1+y⁶) dx = ∫ x⁰ dy
5. The antiderivative of the left side with respect to x is x, and the antiderivative of the right side with respect to y is y/7, assuming y > -1:
6. x + C1 = y/7 + C2
7. Combine the constants of integration:
8. x + C = y/7
9. Solve for y:
10. y = 7x + 7C

This is the general solution to the differential equation.