Lecture 37: Order-Reducible Equations and Higher-Order Linear Differential Equations

1.1 Order-Reducible Equations

Core Idea of Order Reduction

• Core:
• Most higher-order equations cannot be reduced by a standard procedure. Many equations in Section 1.2 have theoretical solutions but no universal elementary method;
• Two common forms that can be reduced and solved systematically are $y^{\prime\prime}=f(x,y^{\prime})$ and $y^{\prime\prime}=f(y,y^{\prime})$;
• Identify the order-reducible form, make the corresponding substitution, and then use the method of separation of variables;
• Note:
• Each integration introduces a constant; use the initial or boundary conditions to determine these constants one by one;

Common Order-Reducible Forms

• 1. $y^{\prime\prime}=f(x)$；
• Example: $y^{\prime\prime}=e^{x},\quad y^{\prime}=e^{x}+C_{1},\quad y=e^{x}+C_{1}x+C_{2}$
• 2. $y^{\prime\prime}=f(x,y^{\prime})$
• Method:
• Set ${y^{\prime}=p,\ y^{\prime\prime}=\frac{dp}{dx}}$;
• This gives the first-order equation $\frac{dp}{dx}=f(x,p)$ in $p$ and $x$, after which the variables can be separated;
• Example:
• $\text{ Differential equations }xy^{\prime\prime}+3y^{\prime}=0\text{ The general definition is }$
• No direct y appears, so the second method is used;
• Getting: $y^{\prime}=p,y^{\prime\prime}=\frac{dp}{dx},x\frac{dp}{dx}+3p=0$ This equation is a separable equation of order 1;
• 3. $y"=f(y,y^{\prime})$
• Method:
• Set: $y^{\prime}=P$, $y^{\prime\prime}=\frac{dp}{dy}P$;
• Become a first-order equation with only y and p and no x, then use the method of separating variables;

1.2 Higher-Order Linear Differential Equations

Summary

• Higher-order linear differential equations;
• Includes:
• Higher-order linear homogeneous differential equations;
• Higher-order linear nonhomogeneous differential equations;
• Features:
• There are theoretical solutions, but no universal solution methods;
• Cases that can be solved:
• Homogeneous linear differential equations with constant coefficients;

Structure of Solutions

• Second-order linear homogeneous differential equation:
• $$y^{\prime\prime}+p(x)y^{\prime}+q(x)y=0$$
• Second-order linear nonhomogeneous differential equation:
• $$y^{\prime\prime}+p(x)y^{\prime}+q(x)y=f(x)$$

#####Theorem: Solutiontoasecondorderlinearhomogeneousdifferentialequation

Description: If $y_1(x)$ and $y_2(x)$ are two linearly independent solutions to a second-order linear homogeneous differential equation, then: $$y=C_{1}y_{1}(x)+C_{2}y_{2}(x)$$

is a general solution to second-order linear homogeneous differential equations;

Explanation

• The number of independent differential equations matches their order;

Testing Linear Independence

• If $\frac{y_{1}(x)}{y_{2}(x)}$ is not constant on the interval, then $y_1$ and $y_2$ are linearly independent;

Theorem: SolutionsOfSecondOrderLinearNonhomogeneousDifferentialEquations

Description: If $y^*$ is a particular solution of a second-order linear nonhomogeneous differential equation and $y_1(x),y_2(x)$ are linearly independent solutions of the corresponding homogeneous equation, then

$$y=C_{1}y_{1}(x)+C_{2}y_{2}(x)+y^{*}(x)$$

is the general solution of the nonhomogeneous equation;

Explanation

• $$\text{general solution of the homogeneous equation}+\text{one particular solution}=\text{general solution of the nonhomogeneous equation}$$

Theorem: RelationshipBetweenNonhomogeneousAndHomogeneousSolutions

Description: If $y^_1(x)$ and $y^_2(x)$ are two particular solutions of the same nonhomogeneous differential equation, then $y^_1(x)-y^_2(x)$ is a solution of the corresponding homogeneous equation;

Theorem: SuperpositionOfNonhomogeneousTerms

Description: If $y^_1(x)$ and $y^_2(x)$ are particular solutions of

$$y^{\prime\prime}+p(x)y^{\prime}+q(x)y=f_{1}(x),\qquad y^{\prime\prime}+p(x)y^{\prime}+q(x)y=f_{2}(x),$$

respectively, then $y^_1(x)+y^_2(x)$ is a particular solution of

$$y^{\prime\prime}+p(x)y^{\prime}+q(x)y=f_1(x)+f_2(x).$$