Differential Equation:
An equation containing the derivatives of one or more dependent variables (unknown functions), with respect to one or more independent variables, is said to be a differential equation (DE).
CLASSIFICATION BY TYPE
i)Ordinary Differential Equation (ODE)
ii)Partial Differential Equation (PDE).
If an equation contains only ordinary derivatives of one or more dependent variables with respect to a single independent variable it is said to be an ordinary differential equation (ODE).
Example: dx / dt + dy / dt =2x +y
An equation involving partial derivatives of one or more dependent variables of two or more independent variables is called a partial differential equation (PDE).
Example: ∂2u / ∂x2 + ∂2u / ∂y2 = 0
Notations
i)Leibniz notation dy / dx, d2y / dx2, d3y / dx3, . . .
ii)Prime notation y`, y``, y```, . . .
iii)Dot notation (flyspeck notation) denotes derivatives with respect to time t.
Sod2s / dt2 = - 32 becomes s¨= -32.
iv)Partial derivatives are often denoted by a subscript notation indicating the independent variables.
For example ∂2u / ∂x2 + ∂2u / ∂y2 = 0 uxx + uyy= 0.
Classification by Order
The order of a differential equation (either ODEor PDE) is the largest derivative present in the differential equation.
It is second-order ordinary differential equation.
CLASSIFICATION BY LINEARITY
- Linear
- Nonlinear
Alinear differential equationis any differential equation that can be written in the following form.
Two properties of a linear ODE are as follows:
The dependent variable y and all its derivatives y`, y``, . . . , y(n)are of the first degree, that is, the power of each term involving y is 1.
The coefficients a0, a1, . . . , anof y, y`, . . . , y(n)depend at most on the independent variable x.
Nonlinear ordinary differential equation is simply one that is not linear.
Nonlinear functions of the dependent variable or its derivatives, such as sin y or ey, cannot appear in a linear equation.
first-order ODE second-orderODE fourth-order ODE
See EXERCISES 1.1 Page#10 (A First Course in DE 9th Edition)
Solution of an ODE
Asolutionto a differential equation on an intervalis any functionwhich satisfies the differential equation in question on the interval.
Separable Equation
See EXERCISES 2.2 Page No. 50
2.3LINEAR EQUATIONS
Homogeneous vs non-homogeneous
When g(x) =0, the linear equation (1) is said to be homogeneous;
otherwise, it is nonhomogeneous.
EXERCISES 2.3 Page No. 61
Degreeof aDifferential Equation:
TheDegreeof adifferential equationis the power of the highest order derivative in theequation.
Order: 3
Degree: 1
Order: 2
Degree: 3
Exact Equation
2nd Method
This method is so easy for Exact DE.
⌠ M dx + ⌠ term N (not containing x)dy = C`
Exact Equation
Q: (2x – y + 1) dx + (2y – x - 1) dy = 0
-1;
Hence and the equation is Exact equation
⌠ (2x – y + 1) dx+ ⌠ (2y - 1) dy = 0
x2 – xy + x + y2 – y = 0
So, required solution is:
x2 + y2 – xy + x – y = 0
SOLUTIONS BY SUBSTITUTIONS
When solution is impossible with Separable Method.
We use SUBSTITUTIONS Method.
Example:
2
Put u = y + x
And ;
Put in question
2
2+ 1 now this is separable eq.
Integrating both sides:
Tan-1u = x + cu = tan (x + c)
we know thatu = y + x
so,y + x = tan (x + c)
Ans:y = tan (x + c) – x
Another Example:
2nd degree equation
We get
Substitute v =
Put in question
Now it is separable eq.
By integrating, we get
2.6A NUMERICAL METHOD
Not important
Another type of problem consists of solving a linear differential equation of order two or greater in which the dependent variable y or its derivatives are specified at different points. A problem such as
is called a boundary-value problem (BVP).where y0 and y1 denote arbitrary constants.
The prescribed values y(a) = y0 and y(b) =y1 are called boundary conditions.
A BVP Can Have Many, One, or No Solutions
has infinitely many solutions.
Def:In calculus differentiation is often denoted by the capital letter D—that is, . The symbol D is called a Differential Operator, because it transforms a differentiable function into another function.
Higher-order derivatives can be expressed in terms of D in a natural manner:
General Form:
In general, we define an nth-order differential operator or polynomial operator to be:
L is a linear operator.
It means that there should be one or more than one constant i.e ≠ 0.
A Second Solution by Reduction of Order
Q:y1 = ex; y`` - y = 0
Solution:
y`` - y = 0(1)
Let y = y1 v
y = ex. v(2)
y` = ex v` + v ex
Y``= (exv`` + v` ex) + (vex + ex v`)
Y``= exv`` + v` ex + vex + ex v`
Y``= exv`` + v` ex + vex + ex v`
Y``= exv`` + 2v` ex + vex
Put these in eq. (1)
y`` - y =0
(ex v`` + 2v` ex + vex) - vex= 0
exv`` + 2v` ex + vex - vex=0
exv`` + 2v` ex=0(3)
Now let w = v`
w` = v``
put in eq. (2)
ex w` + 2w ex=0
ex+2wex=0
ex = -2w ex
= -2ex
Taking Integration on both sides:
ln |w| = -2 ex+ c
Taking antilog:
w= e(-2 ex + c)
w = e-2 ex. eC0
w = e-2 ex.C1
w = C1e-2 ex
we know that w = v`
v` = C1e-2 ex
Taking integration on both sides:
v = + C
C
v = C e-2ex + C
put this in eq. (2)
y = ex. v = ex. (C e-2ex+ C)
y =(Cex e-2ex+ Cex)
y = C ex(e-2ex +1)General Solution
y2 = (C ex e-2ex + C ex)Second Solution
we find two solutions
It follows that the general solution of on this interval is
When m1 =m2, we necessarily obtain only one exponential solution,
Higher-Order Equations
If all the roots of (12) are real and distinct, then the general solution of (1) is
Case ll:
EXERCISES 4.3 Q NO. 1 to 14