Final answer:
To find the solution to the differential equation dz/dt = 8t⁴e⁵ᶻ that passes through the origin, we need to integrate the equation.
Step-by-step explanation:
To find the solution to the differential equation dz/dt = 8t⁴e⁵ᶻ that passes through the origin, we need to integrate the equation.
∫dz/8t⁴e⁵ᶻ = ∫dt
∫e⁻⁵ᶻzdz = ∫8t⁴dt
Using the property of exponential integration, the integral of e⁻⁵ᶻz with respect to z is (1/-5)*(e⁻⁵ᶻz), and the integral of 8t⁴ with respect to t is (2t⁵).
Therefore, the solution to the differential equation is: z = -(1/5)*(e⁻⁵ᶻz) + (2/5)*t⁵ + C, where C is the constant of integration.