# 数学代写|常微分方程代写ordinary differential equation代考|MATH3331

## 数学代写|常微分方程代写ordinary differential equation代考|Linear systems with singularities

Now we want to extend the results from the previous section to linear systems
$$w^{\prime}=A(z) w, \quad w\left(z_0\right)=w_0, \quad z, z_0 \in \Omega \subseteq \mathbb{C},$$
where $A(z)$ is a matrix whose coefficients are analytic in $\Omega$.

As in the real case one can show that one can always extend solutions. However, extensions along different paths might give different solutions in general, as we have seen in example (4.8). These problems do not arise if $\Omega$ is simply connected.

Theorem 4.5. Suppose $w^{\prime}=A(z) w+b(z)$ is linear, where $A: \Omega \rightarrow \mathbb{C}^{n \times n}$ and $b: \Omega \rightarrow \mathbb{C}^n$ are analytic in a simply connected domain $\Omega \subseteq \mathbb{C}$. Then for every $z_0 \in \Omega$ the corresponding initial value problem has a unique solution defined on all of $\Omega$.

In particular, the power series for every solution will converge in the largest disc centered at $z_0$ and contained in $\Omega$.

Proof. If $\Omega$ is a disc centered at $z_0$ the result follows as in Corollary $2.6$. For general $\Omega$, pick $z \in \Omega$ and let $\gamma:[0,1] \rightarrow \Omega$ be a path from $z_0$ to $z$. Around each point $\gamma(t)$ we have a solution in a ball with radius independent of the initial condition and of $t \in[0,1]$. So we can define the value of $w(z)$ by analytic continuation along the path $\gamma$. Since $\Omega$ is simply connected, this value is uniquely defined by the monodromy theorem.

This result has the important consequence that a solution of a linear equation can have singularities (poles, essential singularities, or branch points) only at the points where the coefficients have isolated singularities. That is, the singularities are fixed and do not depend on the initial condition. On the other hand, nonlinear equations will in general have movable singularities, as the simple example
$$w^{\prime}=-w^2$$
whose general solution is
$$w(z)=\frac{1}{z-z_0},$$
shows.

## 数学代写|常微分方程代写ordinary differential equation代考|The Frobenius method

In this section we pursue our investigation of simple singularities. Without loss of generality we will set $z_0=0$. Since we know how a fundamental system looks like from Theorem 4.7, we can make the ansatz
$$W(z)=U(z) z^M, \quad U(z)=\sum_{j=0}^{\infty} U_j z^j, \quad U_0 \neq 0 .$$
Using
$$A(z)=\frac{1}{z} \sum_{j=0}^{\infty} A_j z^j$$
and plugging everything into our differential equation yields the recurrence relation
$$U_j(j+M)=\sum_{k=0}^j A_k U_{j-k}$$
for the coefficients $U_j$. However, since we don’t know $M$, this does not help us much. By (4.77) you could suspect that we just have $M=A_0$ and $U_0=\mathbb{I}$. Indeed, if we assume $\operatorname{det}\left(U_0\right) \neq 0$, we obtain $U_0 M=A_0 U_0$ for $j=0$ and hence $W(z) U_0^{-1}=U(z) U_0^{-1} z^{A_0}$ is of the anticipaled form. Unforimalely, we don’t know that $\operatorname{det}\left(U_0\right) \neq 0$ and, even worse, this is wrong in general (examples will follow).

So let us be less ambitious and look for a single solution first. If $\mu$ is an eigenvalue with corresponding eigenvector $u_0$ of $M$, then
$$w_0(z)=W(z) u_0=z^\mu U(z) u_0$$
is a solution of the form
$$w_0(z)=z^\alpha u_0(z), \quad u_0(z)=\sum_{j=0}^{\infty} u_{0, j} z^j, \quad u_{0,0} \neq 0, \alpha=\mu+m .$$
Here $m \in \mathbb{N}0$ is chosen such that $u_0(0)=u{0,0} \neq 0$. Inserting this ansatz into our differential equation we obtain
$$(\alpha+j) u_{0, j}=\sum_{k=0}^j A_k u_{0, j-k}$$
respectively
$$\left(A_0-\alpha-j\right) u_{0, j}+\sum_{k=1}^j A_k u_{0, j-k}=0$$

# 常微分方程代考

## 数学代写|常微分方程代写ordinary differential equation代考|Linear systems with singularities

$$w^{\prime}=A(z) w, \quad w\left(z_0\right)=w_0, \quad z, z_0 \in \Omega \subseteq \mathbb{C},$$

$$w^{\prime}=-w^2$$

$$w(z)=\frac{1}{z-z_0},$$

## 数学代写|常微分方程代写ordinary differential equation代考|The Frobenius method

$$W(z)=U(z) z^M, \quad U(z)=\sum_{j=0}^{\infty} U_j z^j, \quad U_0 \neq 0 .$$

$$A(z)=\frac{1}{z} \sum_{j=0}^{\infty} A_j z^j$$

$$U_j(j+M)=\sum_{k=0}^j A_k U_{j-k}$$

$W(z) U_0^{-1}=U(z) U_0^{-1} z^{A_0}$ 是预期的形式。我们不知道det $\left(U_0\right) \neq 0$ 而且，更糟糕的是，这通常是错䢔的 (示例如下)。

$$w_0(z)=W(z) u_0=z^\mu U(z) u_0$$

$$w_0(z)=z^\alpha u_0(z), \quad u_0(z)=\sum_{j=0}^{\infty} u_{0, j} z^j, \quad u_{0,0} \neq 0, \alpha=\mu+m .$$

$$(\alpha+j) u_{0, j}=\sum_{k=0}^j A_k u_{0, j-k}$$

$$\left(A_0-\alpha-j\right) u_{0, j}+\sum_{k=1}^j A_k u_{0, j-k}=0$$

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