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# 数学代写|常微分方程代考Ordinary Differential Equations代写|MATH340 Higher-Order Exact and Adjoint Equations

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## 数学代写|常微分方程代考Ordinary Differential Equations代写|Higher-Order Exact and Adjoint Equations

The concept of exactness which was discussed for the first-order DEs in Lecture 3 can be extended to higher-order DEs. The $n$ th-order DE (1.5) is called exact if the function $F\left(x, y, y^{\prime}, \ldots, y^{(n)}\right)$ is a derivative of some differential expression of $(n-1)$ th order, say, $\phi\left(x, y, y^{\prime}, \ldots, y^{(n-1)}\right)$. Thus, in particular, the second-order DE (6.1) is exact if
$$p_0(x) y^{\prime \prime}+p_1(x) y^{\prime}+p_2(x) y=\left(p(x) y^{\prime}+q(x) y\right)^{\prime},$$
where the functions $p(x)$ and $q(x)$ are differentiable in $J$.
Expanding (30.1), we obtain
$$p_0(x) y^{\prime \prime}+p_1(x) y^{\prime}+p_2(x) y=p(x) y^{\prime \prime}+\left(p^{\prime}(x)+q(x)\right) y^{\prime}+q^{\prime}(x) y,$$
and hence it is necessary that $p_0(x)=p(x), p_1(x)=p^{\prime}(x)+q(x)$, and $p_2(x)=q^{\prime}(x)$ for all $x \in J$. These equations in turn imply that
$$p_0^{\prime \prime}(x)-p_1^{\prime}(x)+p_2(x)=0 .$$
Thus, the DE (6.1) is exact if and only if condition (30.2) is satisfied.
Similarly, the second-order nonhomogeneous DE (6.6) is exact if the expression $p_0(x) y^{\prime \prime}+p_1(x) y^{\prime}+p_2(x) y$ is exact, and in such a case (6.6) is
$$\left[p_0(x) y^{\prime}+\left(p_1(x)-p_0^{\prime}(x)\right) y\right]^{\prime}=r(x) .$$
On integrating (30.3), we find
$$p_0(x) y^{\prime}+\left(p_1(x)-p_0^{\prime}(x)\right) y=\int^x r(t) d t+c,$$
which is a first-order linear $\mathrm{DE}$ and can be integrated to find the general solution of (6.6).

## 数学代写|常微分方程代考Ordinary Differential Equations代写|Oscillatory Equations

In this lecture we shall consider the following second-order linear DE
$$\left(p(x) y^{\prime}\right)^{\prime}+q(x) y=0$$
and its special case
$$y^{\prime \prime}+q(x) y=0,$$
where the functions $p, q \in C(J)$, and $p(x)>0$ for all $x \in J$. By a solution of (31.1) we mean a nontrivial function $y \in C^{(1)}(J)$ and $p y^{\prime} \in C^{(1)}(J)$. A solution $y(x)$ of $(31.1)$ is said to be oscillatory if it has no last zero, i.e., if $y\left(x_1\right)=0$, then there exists an $x_2>x_1$ such that $y\left(x_2\right)=0$. Equation (31.1) itself is said to be oscillatory if every solution of (31.1) is oscillatory. A solution $y(x)$ which is not oscillatory is called nonoscillatory. For example, the DE $y^{\prime \prime}+y=0$ is oscillatory, whereas $y^{\prime \prime}-y=0$ is nonoscillatory in $J=[0, \infty)$.

From the practical point of view the following result is fundamental.
Theorem 31.1 (Sturm’s Comparison Theorem). If $\alpha, \beta \in$ $J$ are the consecutive zeros of a nontrivial solution $y(x)$ of $(31.2)$, and if $q_1(x)$ is continuous and $q_1(x) \geq q(x), q_1(x) \not \equiv q(x)$ in $[\alpha, \beta]$, then every nontrivial solution $z(x)$ of the DE
$$z^{\prime \prime}+q_1(x) z=0$$
has a zero in $(\alpha, \beta)$.
Proof. Multiplying (31.2) by $z(x)$ and (31.3) by $y(x)$ and subtracting, we obtain
$$z(x) y^{\prime \prime}(x)-y(x) z^{\prime \prime}(x)+\left(q(x)-q_1(x)\right) y(x) z(x)=0,$$
which is the same as
$$\left(z(x) y^{\prime}(x)-y(x) z^{\prime}(x)\right)^{\prime}+\left(q(x)-q_1(x)\right) y(x) z(x)=0 .$$
Since $y(\alpha)=y(\beta)=0$, an integration yields
$$z(\beta) y^{\prime}(\beta)-z(\alpha) y^{\prime}(\alpha)+\int_\alpha^\beta\left(q(x)-q_1(x)\right) y(x) z(x) d x=0 .$$

# 常微分方程代写

## 数学代写|常微分方程代考Ordinary Differential Equations代写|Higher-Order Exact and Adjoint Equations

$$p_0(x) y^{\prime \prime}+p_1(x) y^{\prime}+p_2(x) y=\left(p(x) y^{\prime}+q(x) y\right)^{\prime}$$

$$p_0(x) y^{\prime \prime}+p_1(x) y^{\prime}+p_2(x) y=p(x) y^{\prime \prime}+\left(p^{\prime}(x)+q(x)\right) y^{\prime}+q^{\prime}(x) y,$$

$$p_0^{\prime \prime}(x)-p_1^{\prime}(x)+p_2(x)=0 .$$

$$\left[p_0(x) y^{\prime}+\left(p_1(x)-p_0^{\prime}(x)\right) y\right]^{\prime}=r(x) .$$

$$p_0(x) y^{\prime}+\left(p_1(x)-p_0^{\prime}(x)\right) y=\int^x r(t) d t+c,$$

## 数学代写常微分方程代考Ordinary Differential Equations代写|Oscillatory Equations

$$\left(p(x) y^{\prime}\right)^{\prime}+q(x) y=0$$

$$y^{\prime \prime}+q(x) y=0,$$

$$z^{\prime \prime}+q_1(x) z=0$$

$$z(x) y^{\prime \prime}(x)-y(x) z^{\prime \prime}(x)+\left(q(x)-q_1(x)\right) y(x) z(x)=0,$$

$$\left(z(x) y^{\prime}(x)-y(x) z^{\prime}(x)\right)^{\prime}+\left(q(x)-q_1(x)\right) y(x) z(x)=0 .$$

$$z(\beta) y^{\prime}(\beta)-z(\alpha) y^{\prime}(\alpha)+\int_\alpha^\beta\left(q(x)-q_1(x)\right) y(x) z(x) d x=0$$

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## MATLAB代写

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