This is a homework question, so I'm essentially asking for hints and not answers due to academic honesty concerns. The course is in real analysis (baby rudin) and it is essentially chapter 9 which should be used to solve this.
The equations are $$f'=xf(x)+x^2, f(0)=0$$ with solution $$ f(x)=e^{\frac{x^2}{2}+c}+ \frac{x^3}{3}-e^c=e^c\left( e^{\frac{x^2}{2}}-1\right)+\frac{x^3}{3}.$$
and $$f(x)=\int_0^x[f(t)+t]\sin(t) dt, f(0)=0$$ which at least for continuous $f$ has solution $$f(x)=e^{-\cos(x)+c}+\sin(x)-x\cos(x)-e^{c}.$$
This is how far I've gotten, but I'm asked to determine uniqueness and explicitly asked to use stuff from CH 9 in Rudin, so I guess something with a contraction/fixed point thm but I can't really figure out what right now.
In the previous part of this exercise we proved that if the jacobian is less than or equal to q, were q<1 its a contraction for a function defined on a convex set. The problem here is that I can't seem to get rid of $e^c$, so I might have not solved it entirely. Suggestions? Or should i assume $c=0$ perhaps?
Thanks in advance!
For existence and uniqueness you are asked to check the assumptions of the Picard-Lindelöf theorem. It is not necessary to repeat the proof of it, which uses the Banach fixed point theorem.
Replace $e^c$ by a multiplicative constant $C$. Which tells that along the way you have missed a sign discussion, or earlier the absolute value signs in $\int\frac1y\,dy=\ln|y|+c$.
The second equation is the Picard iteration form on an ODE that can be obtained by taking the derivative of it. The initial condition there is redundant, since for x=0 the integral automatically has the value 0.
Did you use a modified supremum norm in the proof of Picard-Lindelöf?