Answer:
Explanation:
To solve this initial value problem, we can use the Bernoulli differential equation technique:
Divide both sides of the equation by y^3 to obtain:
3y^(-2) y' + (1 + 8x^7)/y^3 = e^(-x^8)/y^5
Let z = y^(-1), then we can rewrite the equation as:
-3z' + (1 + 8x^7)z = e^(-x^8)
This is now a linear differential equation with an integrating factor of:
μ(x) = e^(∫(1 + 8x^7) dx) = e^(x + 4x^8)
Multiply both sides of the equation by the integrating factor to obtain:
-3(e^(x + 4x^8) z)' = e^(x - x^8)
Integrate both sides with respect to x to get:
-e^(x + 4x^8) z = ∫e^(x - x^8) dx + C
where C is a constant of integration.
Solve for z to get:
z = -1/e^(x + 4x^8) ∫e^(x - x^8) dx - C/e^(x + 4x^8)
Substituting z = y^(-1), we get:
y(x) = (C - ∫e^(x - x^8) dx)/e^(x + 4x^8)
Using the initial condition y(0) = 1, we can solve for C:
y(0) = (C - ∫e^(0 - 0^8) dx)/e^(0 + 4(0^8)) = C/1 = 1
So, C = 1.
Substituting C = 1, we get the solution:
y(x) = (1 - ∫e^(x - x^8) dx)/e^(x + 4x^8)
Therefore, the solution to the initial-value problem is:
y(x) = (1 - ∫e^(x - x^8) dx)/e^(x + 4x^8)
As an explicit function of x, the integral term cannot be simplified any further.