Final answer:
To solve the given IVP using substitution, we can let u = e^9y and differentiate to substitute back into the original equation. Alternatively, we can use the integrating factor method by multiplying both sides by e^(9x) and integrating. The solution for y in terms of x can be obtained from this equation.
Step-by-step explanation:
To solve the IVP y' = e^9y - x, y(0) = 0, we can use substitution. First, we can make a substitution by letting u = e^9y. Then, differentiate both sides of the equation to get du = 9e^9y dy. Substituting these back into the original equation, we get du = 9u - x dx.
To solve this differential equation, we can use the integrating factor method. Multiply both sides of the equation by e^(9x) to get e^(9x) du = 9u e^(9x) - x e^(9x) dx. Now, we can integrate both sides to get e^(9x) u = ∫(9u e^(9x) - x e^(9x)) dx. Solving this integral and rearranging the equation, we get the solution for y in terms of x.