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An ordinary differential equation (ODE) is a differential equation that has one or more functions of one independent variable and its derivatives.
- The phrase ordinary differential equation is used to distinguish itself from the word partial differential equation.
- A partial differential equation consists of unknown functions of two or more variables.
- The main goal of the equation is to determine which function satisfies the equation.
An ordinary differential equation consists of variables and a derivative of the dependent variable with reference to the independent variable.
- Any differential equation consists of either a partial or ordinary function.
- In real life, ordinary differential equations are used to compute the motion or flow of electricity.
- It will help understand the to and fro motion of a pendulum.
Key Terms: Ordinary Differential Equation, Derivatives, Polynomials, Variables, Integration, Partial Differential Equation, Homogeneous Differential Equations, Nonhomogeneous Differential Equations, First-order differential Equation
What is an Ordinary Differential Equations?
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Ordinary differential equations refer to equations that are dependent on a single independent variable. It consists of ordinary derivatives consisting of variables and a derivative of the dependent variable defined with respect to the independent variables.
- The maximum derivatives of the dependent variable determine the order of a differential equation.
- It is commonly referred to as differential equations.
- A differential equation consists of at least one derivative of an unknown function.
- Autonomous ODE, Linear ODE and Non-linear ODE are three types of Ordinary Differential Equations.
- Homogeneous Differential Equations and Non-homogeneous Differential Equations are the other two types of ODE.
- It is used in mathematical models including population growth.
Ordinary differential equations
Example of What is an Ordinary Differential Equations?Example 1: (dy/dx) = cos x Example 2: (d2y/dx2) + k2y = 0 |
Order of Ordinary Differential Equations
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Order of Ordinary Differential Equations refers to the order of the highest derivative that occurs in the equation. An unknown function's derivative, either an ordinary or partial derivative, is included in a differential equation.
- Order is the highest derivative of the dependent variable with respect to the independent variable.
- A partial differential equation involves the derivative of one dependent variable with multiple independent variables.
- Ordinary differential equations are referred to simply as differential equations in this context.
Example of Order of Ordinary Differential EquationsExample: Consider the differential equations dy/dx = ex, (d4y/dx4) + y = 0, (d3y/dx3) + x2(d2y/dx2) = 0. The highest derivatives in these differential equations are of first, fourth, and third order, respectively. |
First Order Differential Equation
The differential equation with a degree of one is known as the first-order differential equation. Within form of derivatives, all linear equations are in the first order.
- It only has a first derivative, including such dy/dx, where x and y are the two variables.
- The equation is written as dy/dx = f (x, y) = y'.
Second-Order Differential Equation
A second-order differential equation is an equation that includes a second-order derivative. d/dx(dy/dx) = d2y/dx2 = f"(x) = y" is how it's written.
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Types of Ordinary Differential Equations
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The different types of Ordinary Differential Equations are as follows:
Autonomous Ordinary Differential Equations
Autonomous Ordinary Differential Equations is a form of differential equation that does not depend upon the value of a variable.
Linear Ordinary Differential Equations
Linear Ordinary Differential Equations are a form of the equation when the equation can be written in terms of derivatives of y. This is further divided into two categories, which are as follows:
Homogenous Differential Equation
If a first-order differential equation may be expressed in any way, it is said to be homogenous. This implies there were no constant terms in linear differential equations.
- Integration of the solution of the homogeneous equation is obtained by omitting the constant term.
- It can yield the solution of any linear ordinary differential equation of any order.
- It can be represented as:
P(x,y)dx + Q(x,y)dy = 0, where P(x,y) and Q(x,y)
- Here P and Q represent homogeneous functions of x and y of the same degree.
Example of Homogenous Differential EquationExample 1: y + x(dy/dx) = 0 is a degree 1 homogeneous differential equation. Example 2: x4 + y4(dy/dx) = 0 is a degree 4 homogeneous differential equation. Example 3: The differential equation xy (dy/dx) + y2+ 2x = 0 is not homogeneous. |
Non-Homogeneous Differential Equations
Non-homogeneous differential equations are differential equations in which the degree of all terms is not the same. A linear differential equation, which really is comparable to the linear equation, is one sort of non-homogeneous differential equation.
- A linear differential equation's typical form is dy/dx + Py = Q, which includes the variable y and its derivatives.
Example of Non-Homogeneous Differential EquationsExample: xy (dy/dx) + y2 + 2x = 0 is not really a homogeneous differential equation, for example. |
Non-linear Ordinary Differential Equations
Non-linear Ordinary Differential Equations are a type of equation that cannot be written in the form of linear combinations of the derivatives of y.
Solution of Ordinary Differential Equation
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The solution of the Ordinary Differential Equation is a form of the curve of the given differential equation. The solution is determined by finding the derivative of the Ordinary Differential Equation.
- It results in the calculation of a number of arbitrary constants, which provide infinite possibilities.
- In the differential equation,
f (x,y,y',….,yn )=0
- A function u:I⊂R→R, where I represents an interval, also called as a solution or integral curve for F.
f (x,u,u',….,un )=0 x ∈1
- Represented, the two solutions u:J⊂R→R and v:I⊂R→R, u is known as an extension of v if I⊂J and
u (x)=v(x) x∈1
- The term "maximal solution" refers to a solution with no extensions.
- On the other hand, term "global solution" refers to a solution that spans all of R.
General Solution
An nth-order equation's general solution is a solution with n arbitrarily independent integration constants. Setting the constants to specific values, which are commonly chosen to satisfy set 'starting conditions or boundary conditions.
- It yields a specific solution from the general solution.
- A solitary solution is one that can't be reached by giving precise values to the general solution's arbitrary constants.
Particular Solution
The phrase "particular solution" refers to any ODE solution (not necessarily satisfying the beginning conditions), which is then combined with the homogeneous solution (a generic solution of the homogeneous ODE) to generate a general solution of the initial ODE.
- In this method, the solution is free from arbitrary constants.
- A particular solution is obtained by substituting values to the arbitrary constants of the general solution
- This will address the method of indeterminate coefficients and parameter variation.
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Singular Solutions
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Singular solutions to ordinary and partial differential equations have been studied since Leibniz's time. Since about the middle of the nineteenth century, it has received considerable attention. Houtain's work on the subject is valuable but underappreciated (1854).
- Darboux (from 1873) was a pioneer in the theory, and he pioneered the geometric explanation of these solutions.
- It was pursued by a number of authors, including Casorati and Cayley.
- The theory of singular solutions of first-order differential equations, as recognised around 1900, is due to the latter (1872).
Things to Remember
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- Ordinary differential equation is a differential equation that distinguish itself from the partial differential equation.
- The maximum derivatives in respect to the selected variable determines the order of a differential equation.
- General and Particular Solution are two Solution of Ordinary Differential Equation.
- Ordinary differential equation involve derivatives of one variable.
- It is used to examine the growth of illnesses in graphs and to study radioactive decay.
Sample Questions
Ques: Find the general solution for the differential equation, y2+x2y'=0. (3 marks)
Ans:y2+x2dydx=0
\({dy\over dx}\)=-y2x2
1/y2dy=-1/x2dx
∫1/y2 dy=-∫1/x2 dx
-1/y=1/x-C
Here we are considering -C in the place of +C, as the constant for integration for a minor convenience, hence
1/x+1/y=C
Ques: Find the PS to the IVP: y'=-6xy and y0=7. (3 marks)
Ans:\({dy\over y}\)=-6x dx
\(\int\)\({dy\over y}\)=\(\int\)-6x dx
in y=-3x2+In C
y=Cexp -3x2
Applying this IC y0=7, ie. y0=cexp 0 =7,
We get the solution as C=7 and y(x)=7exp (-3x2 )
Ques: Find the ordinary differential equations for all straight lines that touch the circle x2+y2=r2. (3 marks)
Ans: Consider y=mx+c as the equation for all the straight lines that touch the circle.
Provided, equation of the circle is x2+y2=r2……(1)
Tangent to the circle, c2=r2(1+m2)
c=r\(\sqrt{1+m^2}\)
We also know that, y=mx+c……2
y=mx+r\(\sqrt{1+m^2}\)……3
y-mx=r\(\sqrt{1+m^2}\)
After differentiating with respect to x, we get \({dy\over dx}\)-m=0
\({dy\over dx}\)=m
Substituting this value in equation (3)
y-(\({dy\over dx}\).x)=r\(\sqrt{1+({dy\over dx})^2}\)
Squaring both sides,
\([y- ({dy \over dx}.x)]^2\)=r\(\sqrt{1+({dy\over dx})^2}\)
\([y- x({dy \over dx})]^2\)=r2\(({1+({dy\over dx}))^2}\)is the hence required ordinary differential equation
Ques: Solve the Ordinary Differential Equation dx/dt=5x-3 for x t. (3 marks)
Ans: \({dx \over5x-3}\)=dt
Integrating on both sides
\(\int\)\({dx \over5x-3}\)=\(\int\)dt
\({1 \over 5}\)log log| 5x-3|=t= C1
5x-3=±exp (5t+5C1)
x=±\({1 \over 5}\)exp (5t+5C1 )+\({3 \over 5}\)
Let C=\({1 \over 5}\)exp(5C1), we are writing the solution as
x(t)=ce5t+\({3 \over 5}\)
Checking whether xt satisfies the ODE
\({dx \over dt}\)=5ce5t
5x-3=5ce5t+3-3=5ce5t
Here, both the equations are equal. Hence the solution is verified.
Ques: Solve y4y'+y'+x2+1=0. (3 marks)
Ans: We know that,
(y4+1) y'=-x2-1
\({y5 \over 5}\)+y=-\({x3 \over 3}\)-x+c
Here, C is an arbitrary constant. This is an implicit solution and hence we cannot easily explicitly solve y in terms of x.
Ques: Solve (x2y2+y) dx+(2x3y-x) dy=0. (3 marks)
Ans: Expanding,
x2y2dx+2x3 y dy+y dx-x dy=0
Here, a is 1 and b is 2. Therefore,
dxy2=y2dx+2xy dy
Dividing the original equation by x2,
y2dx+2xy dy+\({y dx-x dy\over x^2}\)=0
Hence, xy2+yx=C, x≠0
Where, C is an arbitrary constant, y=0, on the domain R stands for a solution to the originally existing equation.
Ques: Solve e-ydy+dx+2x dy=0. (3 marks)
Ans: The equation is linear, with x as a function of y
So,
dx/dy+2x=-e-y
Were, I=e \(\int\)2y dy=e2y
Hence, e2y(dx/dy)+2x e2y=-ey
x.e2y=-ey+C
Were, C is an arbitrary constant. We could solve the explicity for y, but the domain is not that much easy to solve.
Ques: What is Fuchsian theory, and how does it work. (4 marks)
Ans: An innovative method was inspired by two memoirs by Fuchs, which was later refined by Thomé and Frobenius. Beginning in 1869, Collet was a major contributor.
- Bertrand received his technique for integrating a non-linear system in 1868.
- Clebsch (1873) criticised the theory along the same lines as his Abelian integrals theory.
- Clebsch proposed classifying transcendent functions defined by differential equations according to the invariant characteristics of the relating surfaces f = 0 under rational one-to-one transformations.
- As this can be categorised according to the properties of the fundamental curve that remain unchanged under a rational transformation.
Ques: What does it mean to have a system that is integrable. (2 marks)
Ans: Integrability is a feature of certain dynamical systems in mathematics. Informally, an integrable system is a dynamical system with enough conserved quantities, or initial integrals, that its behaviour has considerably fewer degrees of freedom than the dimensions of its phase space; that is, its development is constrained to a submanifold within its phase space.
Ques: What is contact geometry. (4 marks)
Ans: Contact geometry is the science of a geometric structure on smooth manifolds produced by hyperplane distributions in the tangent bundle meeting a 'full non-integrability condition.
- Alternatively, such a distribution can be represented (at least locally) as the kernel of a differential one-form, with the non-integrability condition corresponding to the form's maximum non-degeneracy condition.
- These conditions are the polar opposites of two equivalent conditions for a hyperplane distribution's 'full integrability,' namely, that this is tangent to a codimension one foliation on the manifold, their equivalence is the Frobenius theorem's substance.
Ques: What is the correct order of the ordinary differential equations given (d2y/dx2) + x(dy/dx) + y = 2cosx? (2 marks)
Ans: The highest power of the given derivatives in the required ordinary differential equations is 2 and hence the order of the equation is 2.
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