Wronskian and General Solution of Differential Equations | Mathematical Physics | University Notes | Physics Notes | B.Sc. Physics Notes by Study Buddy Notes

Wronskian and General Solution of Differential Equations

Wronskian and General Solution of Differential Equations | Mathematical Physics | University Notes | Physics Notes | B.Sc. Physics Notes by Study Buddy Notes

The Wronskian is a determinant associated with a set of functions, and it plays a crucial role in determining the linear independence of solutions to differential equations. For a differential equation, the general solution is typically a linear combination of independent solutions, which can be identified using the Wronskian.

1. Definition of the Wronskian

Consider two functions $y_1(x)$ and $y_2(x)$ that are solutions to a second-order linear differential equation. The Wronskian $W(y_1, y_2)(x)$ is defined as:

$$W(y_1, y_2)(x) = \begin{vmatrix} y_1 & y_2 \\ y_1' & y_2' \end{vmatrix} = y_1 y_2' - y_2 y_1'$$

For $n$ functions $y_1, y_2, \ldots, y_n$, the Wronskian $W(y_1,y_2,\dots ,y_n) \; is\; the\; determinant\; of\; the\; matrix\; formed\; by\; these\; functions\; and\; their\; derivatives\; up\; to\; order$$n-1$:

$$W(y_1, y_2, \ldots, y_n) = \det \begin{bmatrix} y_1 & y_2 & \cdots & y_n \\ y_1' & y_2' & \cdots & y_n' \\ \vdots & \vdots & \ddots & \vdots \\ y_1^{(n-1)} & y_2^{(n-1)} & \cdots & y_n^{(n-1)} \end{bmatrix}$$

2. Use of the Wronskian for Linear Independence

The Wronskian is used to test whether a set of functions is linearly independent:

• If $W(y_1,y_2,\dots ,y_n)(x)\mathrm{\ne }\mathrm{0\; for\; some }$$x$ in the interval of interest, the functions $y_1, y_2, \ldots, y_n$ are linearly independent on that interval.
• If $W(y_1, y_2, \ldots, y_n)(x) = 0$ for all $x$ in the interval, the functions are linearly dependent on that interval.

For a second-order differential equation, if $y_1$ and $y_2$ are two solutions with a non-zero Wronskian, they form a fundamental set of solutions, allowing us to construct the general solution.

3. The Wronskian and Differential Equations

For a second-order linear homogeneous differential equation:

$$y^{\mathrm{\prime }\mathrm{\prime }}+p(x)y^{\mathrm{\prime }}+q(x)y=0$$

where $p(x)$ and $q(x)$ are continuous functions, the Wronskian of two solutions $y_1$ and $y_2$ satisfies Liouville's formula:

$$W(y_1,y_2)(x)=W(y_1,y_2)(x_0)e^{-{\int }_{x_0}^{x}p(t)dt}$$

This formula implies that if $W(y_1, y_2)(x_0) \neq 0$ at some point $x_0$, then $W(y_1, y_2)(x) \neq 0$ for all $x$ in the interval, confirming that $y_1$ and $y_2$ are linearly independent on that interval.

4. General Solution of Second-Order Linear Homogeneous Differential Equations

If $y_1(x)$ and $y_2(x)$ are linearly independent solutions to the differential equation, then the general solution is given by:

$$y(x)=C_1{y}_1(x)+C_2{y}_2(x)$$

where $C_1$ and $C_2$ are arbitrary constants. The linear independence of $y_1$ and $y_2$ guarantees that this combination encompasses all possible solutions to the equation.

Example

Consider the equation:

$$y'' - y = 0$$

1. The characteristic equation is:

   $$m^2-1=0$$

2. The roots are $m=\pm \mathrm{1,\; leading\; to\; the\; solutions }$$y_1 = e^x$ and $y_2=e^{-\mathrm{x.}}$

3. Compute the Wronskian of $y_{\mathrm{1<span\; class="katex"><span\; aria-hidden="true"\; class="katex-html"><span\; class="base"><span\; class="mord"><span\; class="msupsub"><span\; class="vlist-t\; vlist-t2"><span\; class="vlist-r"><span\; class="vlist-s"></span></span><span\; class="vlist-r"><span\; class="vlist"></span></span></span></span></span></span></span></span>\; and }}$$y_2$:

   $$W(y_1,y_2)(x)=\mid \begin{array}{cc}{e}^x& e^{-x}\\ e^x& -e^{-x}\end{array}\mid =e^x(-e^{-x})-e^{-x}(e^x)=-2$$

Since $W(y_1, y_2) = -2 \neq 0$, $y_1$ and $y_2$ are linearly independent, and the general solution is:

$$y(x)=C_1{e}^x+C_2{e}^{-x}$$

Summary

• The Wronskian helps verify linear independence.
• For a second-order differential equation, two linearly independent solutions yield the general solution $y(x) = C_1 y_1(x) + C_2 y_2(x)$.
• The Wronskian must be non-zero for a complete, independent solution set.