How to Identify and Solve Singular Solutions with Matrices
A function (x) is known as the singular solution of differential equation F (x, y, y’) = 0 if the uniqueness of the solution is disrupted at each point of the domain of the equation. Geometrically, this implies that more than one integral curve with the common tangent point passes through each point (x₀, y₀).
Note: Sometimes, the weaker singular solution definition is used when the uniqueness of the solution of the differential equation is disrupted only at some point.
A singular solution of a differential equation is not defined by the general integral, that is it cannot be derived from the general solution of any specific value of Constant C.
Singular Solution Example Problem
Suppose the following equation is asked to be solved:
\[(y')^{2} - 4y = 0\]
It can be easily seen that the general solution of the differential equation is given by the function \[y = (x+c)^{2}\].
Graphically, the image is represented by the family of parabolas as shown below:
(Image will be Updated soon)
Besides this, the function y = 0 also satisfies the differential equation. However, this function is not included in the general solution. As more than one integral point passes through each point of the straight line y = 0, the uniqueness of the solution is violated at this point and hence, it is known as a singular solution of the differential equation.
Regular Singular Point
Let us consider the following differential equation
A(x)y’’ + B(x)y’ + C(x)y = 0
If both A and B are polynomials, them regular singular point x₀ is a singular point for which
\[\lim_{x\rightarrow x_{0}}\] (x - x₀) \[\frac{B(x)}{A(x)}\] is infinite and \[\lim_{x\rightarrow x_{0}}\] \[(x - x_{0})^{2}\] \[\frac{C(x)}{A(x)}\] is finite.
For more general function other than polynomial, x₀ is a regular singular point if is considered to be a singular point with
(x - x₀)\[\frac{B(x)}{A(x)}\] and \[(x - x_{0})^{2}\] \[\frac{C(x)}{A(x)}\] are analytic at x = x₀
A singular point that is not a regular singular point is known as an irregular singular point.
Singular Matrix Solution
A matrix is said to be singular if its determinant is equal to 0.
For S example, \[\begin{bmatrix}2 & 6 \\1 & 3 \end{bmatrix}\] is a singular matrix as 2.3 - 6.1 = 0
Note: A singular matrix is non-invertible which implies that its inverse does not exist. Let us understand what inverse does not exist for singular matrices.
The inverse of matrix X is given as:
X' = \[\frac{adjoint(X)}{|X|}\]
In case of singular matrix, |X| = 0
The denominator needs to be 0 in the case of a singular matrix, and that is not defined.
Therefore, the inverse of the singular matrix does not exist.
Solving Singular Matrix
Determine whether the given matrix is singular or not
\[\begin{bmatrix}2 & 4 & 6 \\2 & 0 & 2 \\6 & 8 & 14 \\\end{bmatrix}\]
Solution:
\[\begin{bmatrix}2 & 4 & 6 \\2 & 0 & 2 \\6 & 8 & 14 \\\end{bmatrix}\]
The determinant for the given matrix is calculated as:
2(8 2 - 14 0) - 4 (2 14 - 6 2) + 6(2 8 - 60)
= 2(16 - 0) - 4 (48 - 12) + 6(16 - 0)
= 2(16) - 4(16) + 6(16)
= 32 - 64 + 96
= 0
As the determinant is 0, hence the given matrix is singular.
Singular Solution Example Problems With Solution
1. Find the singular point of the differential equation and classify them as regular or irregular?
(x² - 9)²y’’ + ( x + 3)y’ + 2y = 0
Solution:
Here,
x = 3 is an irregular singular point
x = -3 is a regular singular point.
2. Find the regular singular point of differential equation
(1 - x²)y’’ - 2x y’ + (+ 1)y = 0
Where,
is a real constant
Solution:
As we know:
\[\frac{(x-1) Q(x)}{P(x)}\] = \[\frac{2x}{1+x}\] = \[\frac{(x-1)^{2} R(x)}{P(x)}\] = \[\frac{(x-1) \alpha(\alpha + 1)}{1+x}\]
Furthermore, the limits given below are infinite,
\[\lim_{x \to 1}\] \[\frac{(x-1) Q(x)}{P(x)}\]= 1 , \[\lim_{x \to 1}\] \[\frac{(x-1) R(x)}{P(x)}\]= 0
Therefore, we can conclude that x₀ = 1 is a regular singular point.
FAQs on Singular Solution: Meaning, Examples & Methods
1. What exactly is a singular solution in differential equations?
A singular solution of an ordinary differential equation (ODE) is a unique solution that cannot be derived from the general solution by choosing a specific value for the arbitrary constant. Geometrically, it often represents the envelope of the family of curves described by the general solution, meaning it is a curve that is tangent to every curve in that family.
2. Can you give a simple example of a singular solution?
Certainly. Consider the differential equation (y')² - 4y = 0. Its general solution is y = (x + c)², which represents a family of parabolas. However, the function y = 0 also satisfies the equation. Since you cannot obtain y = 0 by substituting any value for 'c' in the general solution, y = 0 is the singular solution. It acts as the x-axis, which is the envelope for the family of parabolas.
3. What is the standard method for finding a singular solution?
The singular solution is typically found by determining the envelope of the general solution's family of curves. The two primary techniques are:
- The c-discriminant method: You differentiate the general solution with respect to the constant 'c', and then eliminate 'c' between the original general solution and this new derivative equation.
- The p-discriminant method: Letting p = dy/dx, you differentiate the original differential equation with respect to 'p' and then eliminate 'p' between the original ODE and this new derivative.
4. How does a singular solution differ from general and particular solutions?
The key difference is their connection to the arbitrary constant 'c':
- A general solution includes the arbitrary constant 'c' and represents an entire family of solution curves.
- A particular solution is one specific curve from that family, obtained by assigning a value to 'c' (e.g., based on an initial condition).
- A singular solution stands apart from this family. It is not obtained by picking a value for 'c' and often represents a boundary or special case for the entire family of solutions.
5. Is a 'singular solution' in differential equations the same as a 'singular matrix' in algebra?
No, they are completely unrelated concepts from different branches of mathematics. A singular matrix is a matrix with a determinant of zero, meaning it lacks an inverse. A singular solution is a special solution to a differential equation that acts as an envelope. The word 'singular' is used in both contexts to mean 'exceptional' or 'unique', but their technical definitions are entirely different.
6. Why isn't it possible to get a singular solution directly from the general solution?
A singular solution cannot be obtained from the general solution because it doesn't represent a single member of the solution family. Instead, it describes a property of the family as a whole—its envelope. At each point on the singular solution's curve, it is tangent to a different member of the general solution's family. This collective, tangential relationship is why it exists as a separate, unique solution.
