Mathematics. Session Differential Equations - 1 Session Objectives Differential Equation Order and...
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Transcript of Mathematics. Session Differential Equations - 1 Session Objectives Differential Equation Order and...
Mathematics
Session
Differential Equations - 1
Session Objectives
Differential Equation
Order and Degree
Solution of a Differential Equation, General and Particular Solution
Initial Value Problems
Formation of Differential Equations
Class Exercise
Differential Equation
An equation containing an independent variable x,dependent variable y and the differential coefficients of the dependent variable y with respect to independent variable x, i.e.
2
2
dy d y, , …
dx dx
Examples
dy1 = 3xy
dx
2
2
d y2 + 4y = 0
dx
33 2
3 2
d y d y dy3 + + +4y = sinx
dxdx dx
2 24 x dx +y dy = 0
Order of the Differential Equation
The order of a differential equation is the order of the highest order derivative occurring in the differential equation.
2 32 23
2 2
d y dy d y dyExample: = =
dx dxdx dx
The order of the highest order derivative
2
2
d yis 2.
dx
Therefore, order is 2
Degree of the Differential Equation
The degree of a differential equation is the degree of the highest order derivative, when differential coefficients are made free from fractions and radicals.
3322 22 22
2 2
d y dy d y dyExample : + 1+ = 0 = 1+
dx dxdx dx
The degree of the highest order derivative is 2.2
2
d y
dx
Therefore, degree is 2.
Example - 1
Determine the order and degree of the differential
equation:2dy dy
y = x +a 1+dx dx
.
2dy dySolution: We have y = x +a 1+
dx dx
2dy dyy - x = a 1+
dx dx
2 22dy dy
y - x = a 1+dx dx
Solution Cont.
2 22 2 2 2dy dy dy
y - 2xy +x = a +adx dx dx
The order of the highest derivative is 1 and its degree is 2.
dydx
Example - 2
Determine the order and degree of the differential
equation:
324 2
4
d y dyc
dxdx
Solution: We have
3322 24 42
4 4
d y dy d y dy= c + = c +
dx dxdx dx
Here, the order of the highest order is 4 4
4
d y
dx
and, the degree of the highest order is 24
4
d y
dx
Linear and Non-Linear Differential Equation
A differential equation in which the dependent variable y and
its differential coefficients i.e. occur only in the
first degree and are not multiplied together is called a linear differential equation. Otherwise, it is a non-linear differential equation.
2
2
dy d y, , …
dx dx
Example - 3
is a linear differential equation of order 2 and degree 1.
is a non-linear differential equation because the dependent
variable y and its derivative are multiplied together. dydx
2
2
d y dyi - 3 + 7y = 4x
dxdx
dyii y × - 4 = x
dx
Solution of a Differential Equation
The solution of a differential equation is the relation between the variables, not taking the differential coefficients, satisfying the given differential equation and containing as many arbitrary constants as its order is.
For example: y = Acosx - Bsinx
is a solution of the differential equation2
2
d y+4y = 0
dx
General Solution
If the solution of a differential equation of nth order contains n arbitrary constants, the solution is called the general solution.
is the general solution of the differential equation 2
2
d y+y = 0
dx
y Bsin x
is not the general solution as it contains one arbitrary constant.
y = Acosx - Bsinx
Particular Solution
A solution obtained by giving particular values to the arbitrary constants in general solution is called particular solution.
y 3cos x 2sin x
is a particular solution of the differential equation 2
2
d y+y = 0.
dx
Example - 4
3 2Solution: We have y = x +ax +bx +c …(i)
2dy= 3x +2ax +b …(ii) Differentiating i w.r.t. x
dx
2
2
d y= 6x +2a …(iii) Differentiation ii w.r.t. x
dx
3 2
3
3
Verify that y = x +ax +bx+c is a solution of the
d ydifferential equation =6.
dx
Solution Cont.
3
3
d y= 6 Differentiating iii w.r.t x
dx
3
3
d y= 6 is a differential equation ofi .
dx
Initial Value Problems
The problem in which we find the solution of the differential equation that satisfies some prescribed initial conditions, is called initial value problem.
Example - 5
2x x
2
dy d y= e , = e
dx dx
xy = e +1 satisfies the differential equation2
2
d y dy- = 0
dxdx
Show that is the solution of the initial value
problem
xy = e +1
2
2
d y dy- = 0, y 0 = 2, y' 0 = 1
dxdx
xSolution : We have y = e +1
Solution Cont.
0 0
x=0
dyy 0 = e +1 and = e
dx
y 0 = 2 and 'y 0 1
xy = e +1 is the solution of the initial value problem.
x dyy = e +1 and
dxxe
Formation of Differential Equations
y = mx
Assume the family of straightlines represented by
dy= m
dxdy y
dx xdy
x ydx
is a differential equation of the first order.
X
Y
O
ymx m = tan
Formation of Differential Equations
Assume the family of curves represented by
where A and B are arbitrary constants.
y = Acos x +B …(i)
dyA sin x B ... ii
dx [Differentiating (i) w.r.t. x]
2
2
d yand Acos x B
dx [Differentiating (ii) w.r.t. x]
Formation of Differential Equations
2
2
d yy
dx [Using (i)]
2
2
d y+y = 0
dx
is a differential equation of second order
Similarly, by eliminating three arbitrary constants, a differential equation of third order is obtained.
Hence, by eliminating n arbitrary constants, a differential equation of nth order is obtained.
Example - 6
Form the differential equation of the family of curves
a and c being parameters. y = a sin bx + c ,
Solution: We have y = a sin bx + c
is the required differential equation.
2 22 2
2 2
d y d y= -b y + b y = 0
dx dx
[Differentiating w.r.t. x] dy= ab cos bx + c
dx
[Differentiating w.r.t. x] 2
22
d y= -ab sin bx + c
dx
Example - 7
Find the differential equation of the family of all the circles, which passes through the origin and whose centre lies on the y-axis.
If it passes through (0, 0), we get c = 0
2 2x +y +2gx +2ƒ y = 0
This is an equation of a circle with centre (- g, - f) and passing through (0, 0).
Solution: The general equation of a circle is2 2x +y +2gx +2ƒ y +c = 0.
Solution Cont.
Now if centre lies on y-axis, then g = 0.
2 2x +y +2ƒ y = 0 …(i)
This represents the required family of circles.
dyx y
dxƒdy
dx
dy dy2x +2y +2ƒ = 0 Differentiating i w.r.t. x
dx dx
Solution Cont.
2 2
dyx +y
dxx +y - 2y = 0 Substituting the value offdydx
2 2 2dy dyx +y - 2xy - 2y = 0
dx dx
2 2 dyx - y - 2xy = 0
dx
Thank you