Elementary Integration Method

初等积分法 | Elementary Integration Method

This chapter gives primary method of solving special differential functions, which plays a great role in future study.

恰当方程 | Exact Equation

We focus on the symmetrical form

\[ \begin{equation} M(x,y)dx + N(x,y)dy = 0 \label{eq-exact} \end{equation} \]

This can bring us great convenience for digging into one-order ODE because it can gives us both the relation \(y=f(x)\) or \(x=g(y)\).

全微分方程、恰当方程的定义 | Definition of Exact Equation

If there exists a \(\mathit{\varphi}(x, y) \in C^{1}(D)\) such that

\[ d\mathit{\varphi}(x, y) = M(x,y)dx + N(x,y)dy \]

then equation \(\ref{eq-exact}\) is called Exact Equation.

There are some questions to answer:

• How to judge an equation to be exact Equation?
• If so, how to find original function \(\varphi(x, y)\)?
• If not, how to transform it into one exact Equation?

In this pattern, we answer the first two equation and leave the third one after learning LFODE.

方程是恰当的充要条件 | Necessary and Sufficient Condition for exact Equation

Assume \(D\) is a simply connected region, and \(M(x, y)\), \(N(x, y) \in C(D)\) with \(\frac{\partial M}{\partial y}\) and \(\frac{\partial N}{\partial x} \in C^{1}(D)\). Then equation \(\ref{eq-exact}\) is exact Equation if and only if

\[ \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \]

Prove it.

Hints

\(\Rightarrow\) is easy, by using second-order mixed partial derivatives of \(\mathit{\varphi}\).

\(\Leftarrow\). Using Green Formula/Theorem.

\[ \begin{align*} &\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x} \\ \Leftrightarrow \ &\int_{\gamma}P(x, y)dx+Q(x,y)dy = 0 \quad \forall\text{closed loop } \gamma\\ \Leftrightarrow \ &\int_{\gamma}P(x, y)dx+Q(x,y)dy = C \quad \forall\text{curve } \gamma \text{ connecting } (x_0, y_0), (x, y) \\ \Leftrightarrow \ &\exists \mathit{\varphi}(x,y) \in C^{1}(D) \text{ s.t } d\mathit{\varphi}(x, y) = P(x, y)dx+Q(x,y)dy \end{align*} \]

积分因子 | Integral Factor

This part we hope to find \(\mu(x, y)\) so when we multiply it to both sides of equation \(\ref{eq-exact}\)

\[ \mu(x,y)M(x,y)dx + \mu(x,y)N(x,y)dy = 0 \]

there exists \(\mathit{\varphi}(x, y)\) such that

\[ d\mathit{\varphi}(x,y) = \mu(x,y)M(x,y)dx + \mu(x,y)N(x,y)dy \]

Naively, if \(\mathit{\varphi}(x,y)\in C^2\), then

\[ \frac{\partial (\mu M)}{\partial y} = \frac{\partial^2 \mathit{\varphi}}{\partial x\partial y}=\frac{\partial (\mu N)}{\partial x} \]

Theoretically speaking, we have to solve a PDE

\[ \begin{equation} M(x,y)\frac{\partial \mu}{\partial y} - N(x,y)\frac{\partial \mu}{\partial x} = \left(\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}\right)\mu(x,y) \label{eq-pde} \end{equation} \]

However, actually, it is very hard to solve the above PDE. So we focus on some special case like \(\mu(x,y)=\mu(x)\), \(\mu(y)\), \(\mu(x+y)\), \(\mu(xy)\).

Now we have the following theorem to judge whether we can get the above form of integral factors.

方程有特殊类型的积分因子的充要条件 | Necessary and Sufficient Condition of special integral factor of ODE

Equation \(\ref{eq-pde}\) has solution \(\mu(x)\) depending only on \(x\), if and only if

\[ \frac{\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}}{M} \overset{\Delta}{=} G(x) \]

only depends only on \(x\). Then

\[ \mu(x) = e^{\int_{x_0}^{x}G(t)dt} \]

More generally, equation \(\ref{eq-pde}\) has solution \(\mu(\varphi(x,y))\), if and only if

\[ \frac{\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}}{N\frac{\partial \varphi}{\partial x}-M\frac{\partial \varphi}{\partial y}} \overset{\Delta}{=} f(\varphi(x,y)) \]

变量分离方程 | Variable Separation Equation

This chapter we discuss how to solve equation when it is not Exact Equation. The basic idea is, through transformation, we can convert an equation into an exact Equation.

变量分离方程的定义 | Definition of Variable Separation Equation

If there exists \(M_1(x), M_2(y), N_1(x), N_2(y) \in C^1(D)\) such that

\[ M(x, y) =M_1(x)M_2(y), N(x, y) = N_1(x), N_2(y) \]

then we call equation \(\ref{eq-exact}\) Variable Separation Equation.

For this type of equation, we can multiply both sides

\[ \begin{equation} \frac{1}{M_2(y)N_1(x)} \label{eq-sep-factor} \end{equation} \]

equation \(\ref{eq-exact}\) becomes

\[ \frac{M_1(x)}{N_1(x)}dx + \frac{M_2(y)}{N_2(y)}dy = 0 \]

This is an exact equation, and \(\ref{eq-sep-factor}\) is called an Integral Factor of the equation.

we can get its integral

\[ \int_{x_0}^{x}\frac{M_1(t)}{N_1(t)}dt + \int_{y_0}^{y}\frac{M_2(s)}{N_2(s)}ds = c \]

which is easily seen a solution of the original equation.

And don't forget that if there exists \(a_i (i= 1,2,\cdots, m)\) such that \(N_1(a_i) = 0\), or exists \(b_j (j=1,2,\cdots, n)\) such that \(M_2(b_j) = 0\), then of course \(x = a_i, y = b_j\) are also solutions of the original solution.

• 齐次方程 | Homogeneous Equation

The following equation can also be transferred into Variable Separation Equation.

齐次方程的定义 | Definition of Homogeneous Equation

We call \(f(x, y)\) Homogeneous Function of degree \(n\) if

\[ f(tx, ty) = t^n f(x, y) \]

and call equation \(\ref{eq-exact}\) Homogeneous equation if \(M(x, y), N(x, y)\) are Homogeneous Function.

When we let \(y = u x\), then \(dy = xdu + udx\), substitute in the equation and get

\[ \begin{align*} M(x, ux)dx+N(x,ux)(xdu+udx)&=0 \\ \Leftrightarrow [M(x, ux)+N(x,ux)u]dx+N(x,ux)xdu&=0 \end{align*} \]

extract \(x\) out by definition of Homogeneous Equation:

\[ x^n[M(1, u)+N(1,u)u]dx+x^{n+1}N(1,u)du=0 \]

If \(x^{n+1}[M(1, u)+N(1,u)u]\neq 0\), then we divide both sides by this and get

\[ \frac{1}{x}dx + \frac{N(1,u)}{M(1,u)+uN(1,u)}du=0 \]

which is also Variable Separation Equation.

一阶线性微分方程 | Linear First-Order Differential Equation

Now we focus on a really important expression of ODE: Linear First-Order Differential Equation(LFODE):

\[ \begin{equation} \frac{dy}{dx} + p(x)y = q(x) \label{eq-LFODE} \end{equation} \]

• Homogeneous LFODE(H-LFODE)

We let \(q(x)\equiv 0\) in \(\ref{eq-LFODE}\), we get

\[ \begin{equation} \frac{dy}{dx} + p(x)y = 0 \label{eq-H-LFODE} \end{equation} \]

rewrite it into symmetrical form:

\[ p(x)ydx + dy = 0 \]

which is Variable Separation Equation.

So when \(y\neq 0\), multiply both sides \(1/y\) and integrate

\[ \ln{|y|} + \int_{x_0}^{x}p(t)dt = C \]

get \(y\) out of form \(x\):

\[ \begin{align} y &= \pm e^{C}\cdot e^{-\int_{x_0}^{x}p(t)dt} \nonumber \\ &= C_1\cdot e^{-\int_{x_0}^{x}p(t)dt} \end{align} \]

where \(C_1=\pm e^{C} \neq 0\), but we can include trivial solution \(y \equiv 0\) by letting \(C_1 = 0\).

• Non-Homogeneous linear First-Order Differential Equation

we have two ways to get the answer.

Version 1Version 2

Making use of Integral Factors.

To begin with, we convert equation \(\ref{eq-LFODE}\) into symmetrical form:

\[ \begin{equation} (p(x)y-q(x))dx+dy=0 \label{eq-SYM-LFODE} \end{equation} \]

Multiply \(e^{\int_{x_0}^{x}p(t)dt}\) to both sides of equation \(\ref{eq-SYM-LFODE}\):

\[ d\left(\ e^{\int_{x_0}^{x}p(t)dt} y \right) - e^{\int_{x_0}^{x}p(t)dt} q(x)dx = 0 \]

That is,

\[ d\left(\ e^{\int_{x_0}^{x}p(t)dt} y- \int_{x_0}^{x}e^{\int_{x_0}^{s}p(t)dt} q(x)ds \right) = 0 \]

integrate and get

\[ e^{\int_{x_0}^{x}p(t)dt} y - \int_{x_0}^{s}e^{\int_{x_0}^{x}p(t)dt} q(s)ds + C = 0 \]

extract \(y\) out and get:

\[ y = e^{-\int_{x_0}^{x}p(t)dt} \left( C + \int_{x_0}^{s}e^{\int_{x_0}^{x}p(t)dt} q(s)ds \right) \]

Through Variation of Constants.

We make a brave treatment: assume one special solution to equation \(\ref{eq-LFODE}\) is

\[ y = u\cdot e^{-\int_{x_0}^{x}p(t)dt} \]

where \(u\) is a new variable.

Subsititute in equation \(\ref{eq-LFODE}\) and get

\[ \left(p(x) u e^{-\int_{x_0}^{x}p(t)dt} -q(x) \right) dx - u p(x) e^{-\int_{x_0}^{x}p(t)dt} dx + e^{-\int_{x_0}^{x}p(t)dt} du = 0 \]

That is

\[ -q(x) dx + e^{-\int_{x_0}^{x}p(t)dt} du = 0 \]

multiply \(e^{\int_{x_0}^{x}p(t)dt}\) to both sides and integrate

\[ u = \int_{x_0}^{x}e^{\int_{x_0}^{s}p(t)dt}q(s)ds \]

So the special solution is

\[ y = e^{-\int_{x_0}^{x}p(t)dt}\cdot \int_{x_0}^{x}e^{\int_{x_0}^{s}p(t)dt}q(s)ds \]

一阶隐式微分方程 | First-order Implicit Differential Equation

Now we focus on equation

\[ \begin{equation} F(x,y, y') = 0 \label{eq-para} \end{equation} \]

where \(y'\) cannot be explicitly solved out.

• 参数法 | parametric method

If we let \(p = y'\), then equation

\[ \begin{equation} F(x,y,p) = 0 \label{eq-para-p} \end{equation} \]

represents a curved surface in 3-D space.

And we can juggle equation \(\ref{eq-para-p}\) and \(dy=pdx\) to get a curve in the space.