What Is a Singular Solution Definition Formula and Solved Examples
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 in Differential Equations Explained Clearly
1. What is a singular solution in differential equations?
A singular solution of a differential equation is a solution that cannot be obtained by assigning any value to the arbitrary constant in the general solution. In other words, it is an additional solution that is not part of the family of general solutions.
- It often represents the envelope of a family of curves.
- It satisfies the original differential equation.
- It is not derived by simply choosing a constant from the general solution.
2. How do you find the singular solution of a differential equation?
The singular solution is found by eliminating the arbitrary constant between the general solution and its derivative with respect to that constant. The standard steps are:
- Start with the general solution: F(x, y, C) = 0.
- Differentiate partially with respect to C: ∂F/∂C = 0.
- Eliminate C between the two equations.
- The resulting equation gives the singular solution.
3. What is the difference between general solution and singular solution?
The general solution contains arbitrary constants, while the singular solution does not depend on any arbitrary constant and cannot be derived from the general solution.
- General solution: Represents a family of curves (e.g., y = x² + C).
- Singular solution: A specific curve not included in that family.
- The singular solution often forms the envelope of the general family.
4. What is an example of a singular solution?
An example of a singular solution occurs in the differential equation (dy/dx)² = 4y, where the general solution is y = (x + C)² and the singular solution is y = 0.
- The general solution represents a family of parabolas.
- Setting C to any value cannot produce y = 0 for all x.
- Thus, y = 0 is a singular solution.
5. Why is the singular solution called the envelope?
A singular solution is called an envelope because it touches each curve of the general solution family at some point without crossing them. Geometrically:
- Each member of the family is tangent to the singular solution.
- The singular curve forms a boundary to the family of solutions.
- It represents the limiting position of nearby solution curves.
6. Can every differential equation have a singular solution?
No, not every differential equation has a singular solution. A singular solution exists only if:
- The general solution contains an arbitrary constant.
- Eliminating the constant produces an additional valid solution.
- The resulting equation satisfies the original differential equation.
7. How is a singular solution related to Clairaut’s equation?
In Clairaut’s equation, the singular solution is obtained by eliminating the parameter from the general solution and forms the envelope of straight lines. For Clairaut’s equation:
- General form: y = px + f(p)
- Differentiate with respect to p and eliminate p.
- The resulting equation gives the singular solution.
8. How do you verify a singular solution?
A singular solution is verified by substituting it into the original differential equation and checking that it satisfies the equation. The steps are:
- Compute dy/dx for the proposed solution.
- Substitute y and dy/dx into the given equation.
- Confirm both sides are equal.
9. What is the geometric meaning of a singular solution?
Geometrically, a singular solution represents the envelope of a family of curves defined by the general solution. This means:
- It is tangent to each member of the family at some point.
- It acts as a boundary curve.
- It cannot be obtained by choosing any value of the arbitrary constant.
10. What are common mistakes when finding a singular solution?
A common mistake is assuming that any special case of the constant in the general solution is a singular solution. Important points to remember:
- A singular solution must not be obtainable from the general solution.
- You must eliminate the constant using ∂F/∂C = 0.
- Always verify the result in the original differential equation.
