Linear |
More general that includes linear and non-linear:General, including linear and nonlinear |
| $y'+p(t)y=g(t) \tag{*}$ |
$y'=f(t,y) \tag{**}$ |
Theorem 2.4.1If $p(t)$ and $g(t)$ are continuous on an interval, then for every $t_0$ in that interval
and arbitrary value $y_0$ there exists a unique solution $y=\phi(t)$ that satisfies (*) with
the prescribed initial condition $\phi(t_0)=y_0$
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Theorem 2.4.2If functions $f$ and $\frac{\partial f}{\partial y}$ are continuous on an open rectangle in the $t(,y)$-plane, then for every pair $(t_0, y_0)$ in that rectangle
there exists a number $\delta>0$ and a unique function $\phi$ such that $y=\phi(t)$ solves (**) for $t \in (t_0-\delta,t_0+\delta)$ and $\phi(t_0)=y_0$
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Linear equation fits into this framework with $f(t,y)=g(t)-p(t) y$ and $\frac{\partial f}{\partial y}=-p(t)$. So both functions are continuous. So Theorem 2.4.2 is more general, and covers linear equations.
But more general isn't better, as its conclusion is weaker!!! |
Proof of Theorem 2.4.1 consists of proving that $$\phi(t)=\frac{C+\int_{t_0}^t \mu(s)g(s) ds}{\mu(t)}$$ with the integrating factor $\mu(t)=e^{\int p(t)dt}$. See Section 2.1.
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Proof of Theorem 2.4.2 is harder as there is no formula for the solution of a general differential equation of order one!
The proof is based on iterative approximation, which can be converted into a symbolic numerical method. Take $\phi_0(t)=y_0$ (constant). Then compute $\phi_1(t)=y_0+\int_{t_0}^t f(s,\phi_0(s))ds$. Keep iterating
$\phi_n(t)=y_0+\int_{t_0}^t f(s,\phi_{n-1}(s))ds$. See Section 2.8.
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Formulas for solutions |
| Linear
$$
y=\frac{1}{\mu(t)}\int \mu(t) g(t)dt
$$ |
Separable
If $
y'=F(y)G(t)$ then the solution is implicit and satisfies equation $\int \frac{dy}{F(y)}=\int G(t)dt$ |
Exact, Integrating factors $\mu(x)$, $\mu(y)$, $\mu(x y)$ See Section 2.6 |