Final answer:
To solve the given initial value problem, we use a substitution to convert the nonlinear differential equation into a linear one, solve it, and use the initial condition y(1) = 2 to find the particular solution.
Step-by-step explanation:
To solve the initial value problem xy' + y = x⁵y⁴ with x > 0 and y(1) = 2, we follow these steps:
- Identify the knowns: differential equation, initial condition y(1) = 2, and the domain x > 0.
- Rewrite the equation in the form of a Bernoulli's equation which is replaceable with a variable substitution. This involves noticing that the equation can be rewritten as y' + (1/x)y = x⁴y⁴.
- Substitute v = y⁻³ to turn the equation into a linear differential equation of the form dv/dx + (3/x)v = -3x⁴.
- Solve the linear differential equation for v(x).
- Back substitute to find y(x) using the relation v = y⁻³.
- Apply the initial condition y(1) = 2 to determine the constant of integration.
- Substitute the knowns into the derived equation to obtain numerical solutions with units if applicable.
- Check the solution to ensure it is reasonable within the context of the initial condition and domain specified.