Answer:
To solve the given initial value problem, we'll separate variables and then integrate both sides of the equation. Here's the step-by-step solution:
Given: y' = (1 - 13x)y^2
Initial condition: y(0) = -1/5
Let's separate variables by moving all terms involving y to one side and x to the other side:
dy / (y^2) = (1 - 13x) dx
Now, we'll integrate both sides:
∫(1/y^2) dy = ∫(1 - 13x) dx
The left side can be integrated as:
∫(1/y^2) dy = -1/y + C₁
The right side can be integrated as:
∫(1 - 13x) dx = x - (13/2)x^2 + C₂
Now, we have:
-1/y + C₁ = x - (13/2)x^2 + C₂
Next, let's combine the constants:
C = C₁ - C₂
So, we have:
-1/y + C = x - (13/2)x^2
Now, we'll use the initial condition, y(0) = -1/5, to find the value of C:
-1/(-1/5) + C = 0 - (13/2)(0)^2
5 + C = 0
C = -5
Now, we can plug this value of C back into our equation:
-1/y - 5 = x - (13/2)x^2
Let's solve for y:
-1/y = x - (13/2)x^2 + 5
Now, invert both sides:
y = -1 / (x - (13/2)x^2 + 5)
So, the solution to the initial value problem in explicit form is:
y(x) = -1 / (x - (13/2)x^2 + 5)
Explanation: