Differential Equations in 2 variables
$$ \frac{dy}{dx}=f(x,y) $$ where f is defined on a domain D and $(x_{0},y_{0})$ be an interior point of D.
$\phi(x)$ is a solution of this DE IVP (Initial Value Problem) if it satisfies
- $[x,\phi(x)]\in D$ and thus $f[x,\phi(x)]$ is defined for $x\in [\alpha,\beta]$ containing $x_{0}$
- $$\frac{d \phi(x)}{d x}=f(x,\phi(x))$$
- $\phi(x_{0})=y_{0}$
Necessary and sufficient condition #
$\phi(x)$ is a solution of the IVP over $[\alpha,\beta]$ if and only if it satisfies the integral equation: $$\phi(x)= y_{0}+\int_{x_{0}}^{x}f[t,\phi(t)]dt;\forall x\in [\alpha,\beta]$$
necessary part #
If it’s a solution, then it has to satisfy the integral equation.
$$ \begin{align*} \frac{d \phi(x)}{d x}=f(x,\phi(x))\\ \phi(x)=\int_{x_{0}}^{x}f(x,\phi(x))dx+c\\ \text{To satisfy }\phi(x_{0})=y_{0}\\ c=y_{0}\\ \end{align*} $$
sufficient part #
$$ \begin{align*} \phi(x)= y_{0}+\int_{x_{0}}^{x}f[x,\phi(x)]dt\\ \text{Clearly, it satisfies}\\ \frac{d \phi(x)}{d t}=f[x,\phi(x)] \end{align*} $$