I have an inhomogeneous linear differential equations:
$$
y'(x)+g(x)\,y(x)=f(x),\quad y(x_0)=y_0,
$$ and $ x_0 \in I$ and $f,g \in C(I)$.
What can I say about uniqueness and existence of a solution?
Separation of variables can be applied for the homogeneous equation, because $g \in C(I)$ and $ y \in C^1(I) $by definition of solution. Therefore I would only get a solution around $y_0$
For the inhomogeneous equation I can do variation of constants.
How can I argue, that there only one solution on I. Do I have to use Picard and Peano?
I do know,the solution is the sum of the homogenous and inhomogenous solution. For the homogenous one is each linear combination another solution.
How can I put this all together?
Best Answer
We look at the homogeneous equation
$(H) \quad y'(x)+y(x)g(x)=0$.
If $G$ is an anti-derivative of $g$ on $I$, then the general solution of $(H)$ is given by $Ce^{G(x)}$, where $C$ is a constant.
Now let $y_1$ and $y_2$ be solutions of the IVP and put $z:=y_1-y_2.$ Then it is easy to see that $z$ is a solution of $(H)$ with $z(x_0)=0.$
There is a constant $C$ such that $z(x)=Ce^{G(x)}$. From $0=z(x_0)=Ce^{G(x_0)}$ we get $C=0$ and therefore $y_1=y_2.$