1 L11: Homogeneous Linear Equations (5.1)
1.1 n\(^{th}\)-order linear differential equation
An n\(^{th}\)-order linear differential equation has the form:
\[a_n(x)\frac{d^n}{dx^n}y+a_{n-1}(x)\frac{d^{n-1}}{dx^{n-1}}y+\cdots+ a_1(x)\frac{dy}{dx}+a_0(x)y=g(x)\] where \(a_n(x)\neq 0\), and \(g(x)\) is called the forcing function.
If \(\boxed{g(x)=0}\) the diffyq is said to be homogeneous.
A differential equation is defined as linear if it can be expressed in the form:
\(a_n(x) \frac{d^n y}{dx^n} + a_{n-1}(x) \frac{d^{n-1} y}{dx^{n-1}} + \cdots + a_1(x) \frac{dy}{dx} + a_0(x) y = g(x)\)
where:
- \(y\) is the unknown function (dependent variable),
- \(x\) is the independent variable,
- \(a_n(x), a_{n-1}(x), \ldots, a_1(x), a_0(x)\) are functions of \(x\) (which can be constants but must not depend on \(y\) or its derivatives),
- \(g(x)\) is the source term or the nonhomogeneous part of the equation, which depends only on \(x\),
- \(n\) is the order of the differential equation, determined by the highest derivative of \(y\) appearing in the equation.
The key characteristics that define a differential equation as linear are:
Linearity in the Unknown Function and Its Derivatives: The equation must be linear with respect to the unknown function \(y\) and its derivatives. This means that each term of \(y\) and its derivatives must be to the first power (no powers or products of \(y\) and its derivatives) and not be inside functions of \(y\) or its derivatives (e.g., \(\sin(y), e^y, (dy/dx)^2\), etc.).
Coefficients Dependent Only on the Independent Variable: The coefficients \(a_n(x), a_{n-1}(x), \ldots, a_1(x), a_0(x)\) in front of \(y\) and its derivatives can only be functions of the independent variable \(x\) or constants. They must not depend on the unknown function \(y\) or its derivatives.
Additivity and Homogeneity: A linear differential equation satisfies the principles of additivity and homogeneity, meaning that if \(y_1\) and \(y_2\) are solutions, then any linear combination \(c_1y_1 + c_2y_2\) (where \(c_1\) and \(c_2\) are constants) is also a solution. This property is intrinsic to linear equations and is crucial for their solution methods.
If an equation does not meet these criteria, it is considered nonlinear. Nonlinear differential equations can exhibit behaviors and complexities not present in linear equations, such as multiple equilibrium points, limit cycles, and sensitivity to initial conditions, making them more challenging to solve analytically.
1.2 2\(^{nd}\)-order linear differential equation
A linear 2\(^{nd}\)-order differential equation1 has the form:
\[a_2(x)\frac{d^2y}{dx^2}+ a_1(x)\frac{dy}{dx}+a_0(x)y=b(x) \tag{1.1}\]
1.3 Superposition Principle
Theorem 1.1 (Superposition Principle) If \(y_1(x)\) and \(y_2(x)\) are solutions to a homogeneous linear 2\(^{nd}\) order diffyq then so is:
\[y(x)=C_1y_1(x)+C_2y_2(x)\] where \(C_1,C_2\in \mathbb{R}\).
Proof. Since \(y_1(x)\) and \(y_2(x)\) are solutions we have:
\[a_2(x)y_1''(x)+ a_1(x)y_1'(x)+a_0(x)y_1(x)=0\]
\[a_2(x)y_2''(x)+ a_1(x)y_2'(x)+a_0(x)y_2(x)=0\]
\[\text{Let's test: }\boxed{y(x)=C_1y_1(x)+C_2y_2(x)}\]
\[=a_2(x)\left[c_1y_1''(x)+c_2y_2''(x)\right]\] \[+a_1(x)\left[c_1y_1'(x)+c_2y_2'(x)\right]\] \[+a_0(x)\left[c_1y_1(x)+c_2y_2(x)\right]\]
\[=c_1\left[ a_2(x)y_1''+\cdots\right]+c_2\left[ a_2(x)y_2''+\cdots\right]=0\] \[\square \]
\[\frac{d^2 y}{dx^2}+y=0\]
1.3.1 Identifying the Equation as a Linear Second-Order Differential Equation
The given differential equation is:
\[ \frac{d^2 y}{dx^2} + y = 0 \]
This is a second-order differential equation because the highest derivative is the second derivative of \(y\) with respect to \(x\). It is linear because it meets the criteria for linearity:
- The unknown function \(y\) and its derivatives appear to the first power and are not multiplied together or by any function of \(y\).
- The coefficients of \(y\) and its derivatives are functions of \(x\) or constants. In this case, the coefficient of \(\frac{d^2 y}{dx^2}\) is 1 (which is a constant), and the coefficient of \(y\) is also 1 (another constant).
- There are no terms involving functions of \(y\) or its derivatives (like \(\sin(y)\) or \(e^y\)).
Hence, this equation is a linear second-order homogeneous differential equation.
1.3.2 Principle of Superposition
For linear differential equations, the principle of superposition holds. This principle states that if \(y_1(x)\) and \(y_2(x)\) are two solutions to a homogeneous linear differential equation, then any linear combination of these solutions \(c_1y_1(x) + c_2y_2(x)\), where \(c_1\) and \(c_2\) are constants, is also a solution to the differential equation. This property stems directly from the linearity of the equation.
1.3.3 Solving the Differential Equation
To solve the equation \(\frac{d^2 y}{dx^2} + y = 0\), we look for solutions of the form \(y = e^{rx}\), where \(r\) is a constant. Substituting \(y = e^{rx}\) into the differential equation gives:
\[ \frac{d^2}{dx^2}e^{rx} + e^{rx} = 0 \implies r^2e^{rx} + e^{rx} = 0 \]
This simplifies to the characteristic equation:
\[ r^2 + 1 = 0 \]
Solving for \(r\) gives:
\[ r = \pm i \]
Thus, the solutions to the differential equation are based on \(r = i\) and \(r = -i\), which are:
\[ y_1(x) = e^{ix} = \cos(x) + i\sin(x) \] \[ y_2(x) = e^{-ix} = \cos(-x) + i\sin(-x) = \cos(x) - i\sin(x) \]
Since the differential equation is real, we take the real parts as the fundamental set of solutions, which are \(\cos(x)\) and \(\sin(x)\).
1.3.4 General Solution
The general solution to the differential equation, by the principle of superposition, is a linear combination of \(y_1(x)\) and \(y_2(x)\):
\[ y(x) = A\cos(x) + B\sin(x) \]
where \(A\) and \(B\) are constants determined by initial conditions.
1.3.5 Particular Solutions
Two particular solutions could be:
- Setting \(A = 1\) and \(B = 0\), we get \(y(x) = \cos(x)\).
- Setting \(A = 0\) and \(B = 1\), we get \(y(x) = \sin(x)\).
These particular solutions are specific instances of the general solution, illustrating how different initial conditions can lead to different solutions within the framework provided by the general solution.
1.4 Definition: General Solution
If every solution to Equation 1.1 can be written as a linear combination of two solutions \(y_1\) and \(y_2\), then
\[y=C_1y_1(x)+C_2y_2(x)\] is called the general solution to Equation 1.1
1.5 Definition: Linear Independence
A set of \(f^{n}\)s \(\{f_1,f_2,\cdots,f_k\}\) is linearly dependent on an interval I, \(\Leftrightarrow\) there exists \(b_1,b_2,\cdots,b_k\in \mathbb{R}\) not all zero such that
\[b_1f_1+\cdots+b_kf_k=0,\qquad \forall x\in I\] If the set is not linearly dependent is linearly independent.
Theorem 1.2 Suppose \(a_2(x),a_1(x),a_0(x)\in C^0(I)\), and if \(a_2(x)\neq 0\) on \(I\).
Then \[a_2(x)y''+a_1(x)y'+a_0y=0 \tag{1.2}\]
has two linearly independent solutions \(y_1(x)\) and \(y_2(x)\) on \(I\).
In addition, the general solution to Equation 1.2 is given by
\[y(x)=c_1y_1(x)+c_2y_2(x)\]
1.6 Definition: Fundamental set
If every solution to Equation 1.2 can be expressed as a linear combination of \(y_1\) and \(y_2\). Then \(y_1\) and \(y_2\) are said to form a fundamental set.
Theorem 1.3 Suppose \(p\) and \(q\) are continuous on an open interval \((a, b)\), let \(x_0\) be any point in \((a, b)\), and let \(k_0\) and \(k_1\) be arbitrary real numbers. Then the initial value problem \[\boxed{y'' + p(x)y' + q(x)y = 0,\qquad y(x_0) = k_0,\quad y'(x_0) = k_1}\] has a unique solution on \((a, b)\).
1.6.1 Example
\[x^2y''+xy'-4y=0\]
- Verify that \(y_1=x^2\) is a solution on \((-\infty,\infty)\)
- Verify that \(y_2=1/x^2\) is also a solution on \((-\infty,0)\cup(0,\infty)\)
- Verify that if \(c_1\) and \(c_2\) are any constants then \(y=c_1x^2+c_2/x^2\) is a solution on \((-\infty,0)\cup(0,\infty)\).
- Solve the initial value problem \[x^2y''+xy'-4y=0,\quad y(1)=2,\quad y'(1)=0\]
1.7 References
Because the many applications in science and engineering, second order differential equation have historically been the most thoroughly studied class of differential equations.↩︎