14. Solutions to Higher order Linear Equations
Dated: 22-11-2024
Preliminary Theory
In order to solve an \(nth\) order non-homogenous linear differential equation
\[a_{n}(x)\frac{d^{n}y}{dx^{n}}+a_{n-1}(x)\frac{d^{n-1}y}{dx^{n-1}}+\cdot\cdot\cdot+a_{1}(x)\frac{dy}{dx}+a_{0}(x)y=g(x)\]
We first solve for
\[a_{n}(x)\frac{d^{n}y}{dx^{n}}+a_{n-1}(x)\frac{d^{n-1}y}{dx^{n-1}}+\cdot\cdot\cdot+a_{1}(x)\frac{dy}{dx}+a_{0}(x)y=0\]
Super Position
Suppose \(y_1, y_2, \ldots, y_n\) are solutions on the interval \(I\) for
\[a_{n}(x)\frac{d^{n}y}{dx^{n}}+a_{n-1}(x)\frac{d^{n-1}y}{dx^{n-1}}+\cdot\cdot\cdot+a_{1}(x)\frac{dy}{dx}+a_{0}(x)y=0\]
then the following is also a solution.
\[y=c_{1}y_{1}(x)+c_{2}y_{2}(x)+\cdot\cdot\cdot+c_{n}y_{n}(x)\]
where \(c_1, c_2, c_n\) are arbitrary constants.
Notes
- If \(y_1(x)\) is a solution to
homogeneous linear differential equation then \(y = cy_1(x)\) is also a solution.
Homogeneous linear differential equations always possess trivial solution \(y = 0\).- The super position principle does not holds for non
linear differential equations.
Example
The functions
\[y_1 = e^x\]
\[y_2 = c^{2x}\]
\[y_3 = e^{3x}\]
satisfy the homogeneous differential equation
\[\frac{d^{3}y}{dx^{3}}-6\frac{d^{2}y}{dx^{2}}+11\frac{dy}{dx}-6y=0\]
on \((- \infty, \infty)\).
This \(y_1, y_2, y_3\) are all solutions.
Now suppose
\[y=c_{1}e^{x}+c_{2}e^{2x}+c_{3}e^{3x}\]
Then
\[\frac{dy}{dx}=c_{1}e^{x}+2c_{2}e^{2x}+3c_{3}e^{3x}\]
\[\frac{d^{2}y}{dx^{2}}=c_{1}e^{x}+4c_{2}e^{2x}+9c_{3}e^{3x}\]
\[\frac{d^{3}y}{dx^{3}}=c_{1}e^{x}+8c_{2}e^{2x}+27c_{3}e^{3x}\]
Therefore
\[\frac{d^{3}y}{dx^{3}}-6\frac{d^{2}y}{dx^{2}}+11\frac{dy}{dx}-6y=c_{1}(e^{x}-6e^{x}+11e^{x}-6e^{x})+c_{2}(8e^{2x}-24e^{2x}+22e^{2x}-6e^{2x}) + c_3(27e^{3x} - 54e^{3x} + 33e^{3x} - 6e^{3x})\]
\[=c_{1}(12-12)e^{x}+c_{2}(30-30)e^{2x}+c_{3}(60-60)e^{3x}\]
\(\(= 0\)\)
Therefore,
\[y=c_{1}e^{x}+c_{2}e^{2x}+c_{3}e^{3x}\]
is also a solution.
The Wronskian
Suppose that \(y_1, y_2\) are solutions on interval \(I\) of the following.
\[a_{2}\frac{d^{2}y}{dx^{2}}+a_{1}\frac{dy}{dx}+a_{0}y=0\]
Then either
\[W(y_1, y_2) = 0, \forall x \in I\]
or
\[W(y_1, y_2) \ne 0, \forall x \in I\]
To verify this, we write
\[\frac{d^{2}y}{dx^{2}}+P\frac{dy}{dx}+Qy=0\]
\[W(y_{1},y_{2})=\begin{vmatrix}y_{1}&y_{2}\\ y_{1}^{\prime}&y_{2}^{\prime}\end{vmatrix}=y_{1}y_{2}^{\prime}-y_{1}^{\prime}y_{2}\]
\[\implies \frac{dW}{dx} = y_1y_2^{\prime\prime} - y_1^{\prime\prime}y_2\]
We the re-write the equation in the following form
\[y_{1}^{\prime\prime}+Py_{1}^{\prime}+Qy_{1}=0\]
\[y_{2}^{\prime\prime}+Py_{2}^{\prime}+Qy_{2}=0\]
Multiplying first equation by \(y_2\) and second equation by \(y_1\), we have
\[y_{1}^{\prime\prime}y_{2}+Py_{1}^{\prime}y_{2}+Qy_{1}y_{2}=0\]
\[y_{1}y_{2}^{\prime\prime}+Py_{1}y_{2}^{\prime}+Qy_{1}y_{2}=0\]
Subtracting both, we have
\[(y_{1}y_{2}^{\prime\prime}-y_{2}y_{1}^{\prime\prime})+P(y_{1}y_{2}^{\prime}-y_{1}^{\prime}y_{2})=0\]
Which can be re-written as
\[\frac {dW}{dx} + PW = 0\]
And its solution is
\[W=ce^{-\int Pdx}\]
Therefore
- If \(c \ne 0\) then \(W(y_1, _2) \ne 0, \quad \forall x \in I\).
- If \(c = 0\) then \(W(y_1, y_2) = 0, \quad \forall x \in I\).
In General
If \(y_1, y_2, \ldots, y_n\) are solutions on interval \(I\) for
\[a_{n}(x)\frac{d^{n}y}{dx^{n}}+a_{n-1}(x)\frac{d^{n-1}y}{dx^{n-1}}+\cdot\cdot\cdot+a_{1}(x)\frac{dy}{dx}+a_{0}(x)y=0\]
Then
\[W(y_{1},y_{2},…,y_{n})=0, \forall x\in I\]
or
\[W(y_{1},y_{2},…,y_{n})\neq 0, \forall x\in I\]
Fundamental Set of Solutions
A set \(\{y_1, y_2, \ldots, y_n\}\) of \(n\) linearly independent solutions on interval \(I\) for
\[a_{n}(x)\frac{d^{n}y}{dx^{n}}+a_{n-1}(x)\frac{d^{n-1}y}{dx^{n-1}}+\cdot\cdot\cdot+a_{1}(x)\frac{dy}{dx}+a_{0}(x)y=0\]
is called fundamental set of solutions on the interval \(I\).
Existence of Fundamental Set of Solutions
There always exists the fundamental set of solutions for equations of the form
\[a_{n}(x)\frac{d^{n}y}{dx^{n}}+a_{n-1}(x)\frac{d^{n-1}y}{dx^{n-1}}+\cdot\cdot\cdot+a_{1}(x)\frac{dy}{dx}+a_{0}(x)y=0\]
on interval \(I\).
General Solution for Homogeneous Equations
Suppose that
\[\{y_1, y_2, \ldots, y_n\}\]
is a fundamental set of solutions on the interval \(I\) for the equations
\[a_{n}(x)\frac{d^{n}y}{dx^{n}}+a_{n-1}(x)\frac{d^{n-1}y}{dx^{n-1}}+\cdot\cdot\cdot+a_{1}(x)\frac{dy}{dx}+a_{0}(x)y=0\]
Then the general solution on the interval \(I\) is defined to be
\[y=c_{1}y_{1}(x)+c_{2}y_{2}(x)+\ldots+c_{n}y_{n}(x)\]
Where \(c_1, c_2, \ldots, c_n\) are arbitrary constants.
Example
Consider the equation
\[\frac{d^{3}y}{dx^{3}}-6\frac{d^{2}y}{dx^{2}}+11\frac{dy}{dx}-6y=0\]
and suppose that
\[y_1 = e^x\]
\[y_2 = e^{2x}\]
\[y_3 = e^{3x}\]
Then
\[\frac{dy_{1}}{dx}=e^{x}=\frac{d^{2}y_{1}}{dx^{2}}=\frac{d^{3}y_{1}}{dx^{3}}\]
Therefore
\[\frac{d^{3}y_{1}}{dx^{3}}-6\frac{d^{2}y_{1}}{dx^{2}}+11\frac{dy_{1}}{dx}-6y_{1}=12e^{x}-12e^{x}=0\]
Now for \(x \in \mathbb R\).
\[W(e^{x},e^{2x},e^{3x})=\begin{vmatrix}e^{x}&e^{2x}&e^{3x}\\ e^{x}&2e^{2x}&3e^{3x}\\ e^{x}&4e^{2x}&9e^{3x}\end{vmatrix}=2e^{6x}\neq0\forall x\in I\]
Therefore, \(y_1, y_2\) and \(y_n\) form a fundamental solution of the differential equation on \((- \infty, \infty)\).
\[y = c_1e^{x} + c_2e^{2x} + c_3e^{3x}\]
Non Homogeneous Equations
A function \(y_p\) that satisfied the non-homogeneous differential equation.
\[a_{n}(x)\frac{d^{n}y}{dx^{n}}+a_{n-1}(x)\frac{d^{n-1}y}{dx^{n-1}}+\cdot\cdot\cdot+a_{1}(x)\frac{dy}{dx}+a_{0}(x)y=g(x)\]
and is free of parameters is called the particular solution of differential equation.
Example
Suppose that
\[y_p = 3\]
Then
\[y_p^{\prime\prime} = 0\]
So that
\[y_p^{\prime\prime} + 9 y_p = 0 + 9(3)\]
\[= 27\]
Therefore
\[y_p = 3\]
is the particular solution of the differential equation
\[y_p^{\prime\prime} + 9y_p = 27\]
Complementary Function
The general solution
\[y=c_{1}y_{1}+c_{2}y_{2}+\ldots+c_{n}y_{n}\]
of the homogeneous linear differential equation
\[a_{n}(x)\frac{d^{n}y}{dx^{n}}+a_{n-1}(x)\frac{d^{n-1}y}{dx^{n-1}}+\ldots+a_{1}(x)\frac{dy}{dx}+a_{0}(x)y=0\]
is known as the complmentary function for the non-homogeneous linear differential equation.
\[a_{n}(x)\frac{d^{n}y}{dx^{n}}+a_{n-1}(x)\frac{d^{n-1}y}{dx^{n-1}}+\ldots+a_{1}(x)\frac{dy}{dx}+a_{0}(x)y=g(x)\]
General Solution of Non-homogeneous Equations
\[\text{general solution} = \text{complementary solution} + \text{any particular solution}\]
Example
Suppose that
\[y_p = - \frac {11}{12} - \frac 1 2 x\]
Then
\[y_p^\prime = - \frac 1 2, y_p^{\prime\prime} = y_p^{\prime\prime\prime} = 0\]
\[\therefore \frac{d^{3}y_{p}}{dx^{3}}-6\frac{d^{2}y_{p}}{dx^{2}}+11\frac{dy_{p}}{dx}-6y_{p}=0-0-\frac{11}{2}+\frac{11}{2}+3x=3x\]
Therefore, \(y_p\) is the solution of non-homogeneous linear equation above.
Now consider
\[y_{c}=c_{1}e^{x}+c_{2}e^{2x}+c_{3}e^{3x}\]
Therefore
\[\frac{dy_{c}}{dx}=c_{1}e^{x}+2c_{2}e^{2x}+3c_{3}e^{3x}\]
\[\frac{d^{2}y_{c}}{dx^{2}}=c_{1}e^{x}+4c_{2}e^{2x}+9c_{3}e^{3x}\]
\[\frac{d^{3}y_{c}}{dx^{3}}=c_{1}e^{x}+8c_{2}e^{2x}+27c_{3}e^{3x}\]
\[\frac{d^{3}y_{c}}{dx^{3}}-6\frac{d^{2}y_{c}}{dx^{2}}+11\frac{dy_{c}}{dx}-6y_{c} =c_{1}e^{x}+8c_{2}e^{2x}+27c_{3}e^{3x}-6(c_{1}e^{x}+4c_{2}e^{2x}+9c_{3}e^{3x}) +11(c_{1}e^{x}+2c_{2}e^{2x}+3c_{3}e^{3x})-6(c_{1}e^{x}+c_{2}e^{2x}+c_{3}e^{3x})\]
\[=12c_{1}e^{x}-12c_{1}e^{x}+30c_{2}e^{2x}-30c_{2}e^{2x}+60c_{3}e^{3x}-60c_{3}e^{3x}\]
\[=0\]
Thus, \(y_c\) is the general solution of associated homogeneous differential equation.
\[\frac{d^{3}y}{dx^{3}}-6\frac{d^{2}y}{dx^{2}}+11\frac{dy}{dx}-6y=0\]
Hence the general solution for non-homogeneous equation is
\[y=y_{c}+y_{p}=c_{1}e^{x}+c_{2}e^{2x}+c_{3}e^{3x}-\frac{11}{12}-\frac{1}{2}x\]
Super Position Principle for Non-homogeneous Equations
Suppose that
\[y_{p_1}, y_{p_2}, \ldots, y_{p_k}\]
denotes the particular solutions of the \(k\) differential equation.
\[a_{n}(x)y^{(n)}+a_{n-1}(x)y^{(n-1)}+\cdot\cdot\cdot+a_{1}(x)y^{\prime}+a_{0}(x)y=g_{i}(x)\]
where \(i = 1, 2, \ldots, k\) on an interval \(I\).
Then
\[y_p = y_{p_1}(x) + y_{p_2}(x) + \cdots + y_{p_k}(x)\]
is a particular solution for
\[a_{n}(x)y^{(n)}+a_{n-1}(x)y^{(n-1)}+\cdot\cdot\cdot+a_{1}(x)y^{\prime}+a_{0}(x)y=g_{1}(x)+g_{2}(x)+\cdot\cdot\cdot+g_{k}(x)\]
Example
Consider the differential equation
\[y^{\prime\prime}-3y^{\prime}+4y=-16x^{2}+24x-8+2e^{2x}+2xe^{x}-e^{x}\]
Suppose that
\[y_{p_1} = - 4x^2\]
\[y_{p_2} = e^{2x}\]
\[y_{p_3} = xe^x\]
Then
\[y_{p_1}^{\prime\prime}-3y_{p_1}^{\prime}+4y_{p_1}=-8+24x-16x^{2}\]
Therefore
\[y_{p_1}=-4x^{2}\]
is a particular solution of the non homogeneous differential equation
\[y^{\prime\prime}-3y^{\prime}+4y=-16x^{2}+24x-8\]
\[y_{p_2} = e^{2x}\]
\[y_{p_3} = xe^x\]
are particular solutions for
\[y^{\prime\prime}-3y^{\prime}+4y=2e^{2x}\]
and
\[y^{\prime\prime}-3y^{\prime}+4y=2xe^{x}-e^{x}\]
respectively.
Hence,
\[y=y_{p_1}+y_{p_2}+y_{p_3}=-4x^{2}+e^{2x}+xe^{x}\]
is the solution of the differential equation
\[y^{\prime\prime}-3y^{\prime}+4y=-16x^{2}+24x-8+2e^{2x}+2xe^{x}-e^{x}\]
References
Read more about notations and symbols.