Lecture 36: First-Order Linear Differential Equations

1.1 Linear Equations

#####Definition: Linearequation

Description: the unknown function $y=y(x)$ and the derivative of the unknown function $y^{\prime}$ are both linear and therefore called linear;

Standard format: $$y^{\prime}+P(x)y=Q(x)$$

Supplement: If y in the expression is second or even third order, but x is only of one order, you can consider swapping x and y and solving using $\frac{dx}{dy}$;

#####Theorem: Ageneralsolutionformulaforlinearequations

description: $$y=e^{-\int p(x)dx}\left[\int Q(x)e^{\int p(x)dx}dx+C\right]$$

Explanation

• Concept:
• It must be written in the standard form of a linear equation before it can be incorporated into the general solution formula;
• Note:
• 1. For indefinite integrals of $p(x)$ and $Q(x)$, there is no need to take any constant $C$;
• 2. For position $p(x)$, if the following form appears, no absolute value is needed: $\int\frac{1}{x}dx=\ln x$;
• Method:
• Step 1: Organize the differential equation into linear equations;
• Step 2: Import the general solution formula to find the general solution;

1.2 Bernoulli's Equation

1.2.1 Basic Concepts

#####Theorem: Bernoulliequation

description: Based on linear differential equations, multiply $y^{1-\alpha}=u)$ to the right of $Q(x)$:

$$y^{\prime}+p (x) y=Q (x) y^{\alpha} \quad\quad\quad\quad\quad\quad(\alpha\neq1)\quad(y^{1-\alpha}=u)$$

Explanation

• Solution:
• Let $y$ be $y^{1-\alpha}=u$ so that the equation becomes a first-order linear differential equation;
• Method:
• If $\alpha$ equals 0, then it is a linear equation;
• If $\alpha$ equals 1, then separable variables can be used directly;
• Therefore: The current discussion assumes that alpha and are not equal to 0, nor equal to 1;
• Approach:
• Think about how to convert into linear equations;

Handling Methods

• Known: $y^{\prime}+p (x) y=Q (x) y^\alpha$
• Step one
• Divide $y^\alpha$ to the left of the equation, divide by y, and you get:
• $y^{1-\alpha}$
• Step 2: Set z
• Ling $y^{1-\alpha}=z$
• Step three
• Transform into a linear equation

1.2.2 Example Problems

Example Question: $\frac{dy}{dx}+\frac yx=a(\ln x)y^2$

• Analysis
• Analysis
• First, divide y squared by the past:
• $\frac1{y^2}\frac{\mathrm{d}y}{\mathrm{d}x}+\frac1x\frac1y=a\ln x$
• $-\frac{1}{y^{2}}\frac{dy}{dx}=\frac{dt}{dx}$
• Then let y be the negative power of z
• $-\frac{\mathrm{d}z}{\mathrm{d}x}+\frac1xz=\mathrm{alnx}$
• Differentiating by respect to x:
• Question Type: Bernoulliequation

1.3 Total Differential Equations

1.3.1 Basic Concepts

#####Definition: Wholedifferentialequations

description: $$dF(x,y)=P(x,y)dx+Q(x,y)dy=0$$

Explanation

• Concept:
• If $P(x,y)dx+Q(x,y)dy=0$ is the differential of a function $F(x,y)$, then the equation is a total differential equation;
• Judgment:
• If the partial derivatives obtained for $P(y)$ and $Q(x)$ are equal:
• $$\frac{\partial P}{\partial y}=\frac{\partial Q}{\partial x}$$
• Then the current equation is a fully differential equation;
• Solution:
• 1. Partial integral;
• 2. Complete the total differential;
• 3. Line integral;

Explanation

• The differential of a multivariable function is the total differential;
• The differential in a single-variable function only considers the change caused by x as an independent variable:
• $dy=A\Delta x$
• In a binary function, the change caused by the two variables x and y must be expressed:
• $dz=Adx+Bdy$
• Partial increments and full increments
• Marginal increment: $\begin{aligned}\Delta z_{x}=f\left(x_{0}+\Delta x,y_{0}\right)-f\left(x_{0},y_{0}\right)\\\Delta z_{y}=f\left(x_{0},y_{0}+\Delta y\right)-f\left(x_0,y_{0}\right)\end{aligned}$
• Full increment: $\Delta z=f(x_{0}+\Delta x,y_{0}+\Delta y)-f(x_{0},y_{0})$

Two Problems of Total Differentiation

• A binary function has two independent variables: x and y
• The definition of the single-variable derivative is conceptually very similar to that of the binary differential;
• Two problems of differentiation
• 1. Is it differentiable?
• (Differentiation in one variable) Since differentiability is differentiable, the properties of derivatives can be used to judge;
• Does binary differential draw such a conclusion?
• 2. How to calculate differentiation
• (Differential of a single variable) Since $dy=f^{\prime}(x)dx$, we can use derivatives to find the part of A in $dz=Adx$;
• Reduce the calculation of differentiation to the calculation of derivatives

1.3.2 Properties of Total Differentials

#####Theorem: Anecessaryconditionformultivariatefunctionstobedifferentiable

Description: If function $z=f(x,y)$ is differentiable at point $(x,y)$, then the partial derivatives $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$ of the function at points (x, y) must exist, and: $dz=\frac{\partial z}{\partial x}\Delta x+\frac{\partial z}{\partial y}\Delta y$

Note: Among them

1. $\frac{\partial z}{\partial x}$ is the A in $dz=Adx+Bdy$;

2. $\frac{\partial z}{\partial y}$ is the B in $dz=Adx+Bdy$;