Get to know about Differential Equations and Partial Differential Equations in Real Life

1. Submitted By –
Akshay Tripathi
B.Tech,II Sem
Submitted By –
Dr. Sona Raj
fessor,Mathematics

2. DIFFERENTIAL EQUATIONS
Mathematics

3.  Linear Differential Equations
 Differential Equations of Second Order
 Applications of Differential Equations

4. The standard form of a linear differential equation of
first order and first degree is
where P and Q are the functions of x, or constants.
dy
+Py = Q
dx
   
 
x
2
dy dy
Examples: 1 +2y = 6e ; 2 +ytanx = cosx ;
dx dx
ydx
3 + = y etc.
dy x

5. dy
Rule for solving +Py = Q
dx
where P and Q are the
functions of x, or constants.
Pdx
Integratingfactor(I..F.)=e
   The solution is y I.F. = Q×(IF) dx+C


6. xdy
Solve the differential equation +2y = 6e .
dx
xdy
Solution: The given differential equation is +2y = 6e .
dx
It is a linear equation of the form dy
+Py =Q
dxx
Here P =2 and Q = 6e
Pdx 2dx 2x
.F.= e = e = eI  
The solution is given by    y I.F. = Q I.F. dx +C

 2x x 2x
y e = 6e ×e dx+C
  2x 2xd dy
Note: ye =e +2y
dx dx
  
  
  

7. 2x 3x
ye = 6e dx +C

2x 3x
ye = 6 e dx +C

3x
2x e
ye = 6× +C
3

2x 3x
ye = 2e +C
3x
2x
2e +C
y =
e

x -2x
y =2e +C×e is the required solution.

8. Solve the following differential equation:
 
-1
2 tan xdy
(1+ x ) + y = e CBSE 2002
dx
Solution: The given differential equation is
It is a linear differential equation of the form dy
+Py = Q
dx
-1
tan x
2 2
1 e
Here, P = and Q =
1+x 1+x
-1
-1 tan x
2 tan x
2 2
dy dy 1 e
(1+x ) +y = e + .y =
dx dx 1+x 1+x


9. The solution is given by
-1
-1 2tan x
tan x e
ye = +C
2

-1 -1
tan x 2tan x
2ye = e +C is the required solution.
-12
1
dx
Pdx tan x1+xI.F = e = e = e


y × I.F. = Q × I.F.dx + C
-1
-1 -1 tan x
tan x tan x
2
e
y×e = e × dx+C
1+x
 

10.  
dy
Solve the differential equation + secx y = tanx.
dx
 
dy
Solution: The given differential equation is + secx y = tanx.
dx
It is a linear differential equation of the form
dy
+Px = Q
dx
Here P = secx and Q= tanx
ePdx secx dx log secx + tanx
I.F.= e = e = e = secx + tanx 

11.    y × IF = Q ×IF dx+C

   y secx + tanx = tanx secx + tanx dx +C

The solution is given by
  2
y secx + tanx = secxtanx dx + tan x dx +C
 
   2
y secx+ tanx = secx+ sec x -1 dx+C

 y secx+tanx = secx+ tanx - x+C

12. dx
Rule for solving +Px = Q
dy
Pdy
Integratingfactor (I.F.)=e
   The solution is x I.F. = Q×(IF) dy +C

where P and Q are the functions of y, or constants.

13. 1) Money:
a) If you can modify a vehicle’s geometry to significantly
reduce turbulent drag (race car, commercial airplane…)
b) Modeling financial instruments (derivatives…)
2) Scientific curiosity:
a) Model’s of poorly understood physical phenomena
(turbulence…)
b) Astrophysical models, solar models…

14. 1)Engineering Applications:
a)Structural modeling
b)Electromagnetics, acoustics, fluid dynamics…
2)Environment:
a)Modeling environmental impact of those pesky
greenhouse gases
b)Modeling weather to avoid damage or to predict crop
performance
c)Predicting earthquakes, volcanic eruptions, tsunami
(all belong in the “Money” section too?.
3)Defense:
a)Designing materials and profiles for stealth aircraft
b)Nuclear weapon stockpile stewardship

15. Thank you