Existence of successive Picard approximations, $y'=3y +1, \enspace y(0)=2$

Question

I'm trying to show that the successive approximations for $$y'=3y +1, \enspace y(0)=2$$

exist for all real $x$. I think the general formula for approximation $\phi_k$ is

$$\phi_k=2+7x+\frac{21}{2}x^2+\frac{21}{2}x^3+\cdots+\frac{k-1}{k}\bigg(\frac{21}{2}x^k\biggr)$$

which I have inferred by computing some of the approximations, so it is not a real solution. My book says to use induction to show the approximations exist. I'm confused about what this means. What would be my inductive statement in this case?

Answer 1

I think I came up with a way to prove this. There is no need to derive a general formula for $\phi_k$. Proceed to proof by induction. The base case $$\phi_1 = 2$$

clearly exists ( is defined ) for all $x$. Now suppose that $$\phi_k = 2 + \int_0^{x}3(\phi_{k-1}(t))+1dt$$

exists for all $x$. By Picard ( successive approximation method ) we have that $$\phi_{k+1} = 2+ \int_0^{x} 3\bigg(2+\int_0^{x}3(\phi_{k-1}(t))+1dt\bigg)+1 dt$$

exists for all $x$. Why? Because it will be a polynomial of degree $k+1$, and as such is continuous everywhere and defined for all $x$. Thus all approximations exists by induction? Does this make sense?