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1. BIET
NAME : SAMIR UDDIN
UNIVERSITY ROLL : 11800722069
UNIVERSITY REGISTRATION NO : 221180120383
DEPARTMENT : MECHANICAL ENGINEERING
SEMESTER : 5TH
SUBJECT NAME : SOLID MECHANICS
SUBJECT CODE : PC-ME502

3. DERIVATION OF CAUCHY RELATIONS AND
EQUIBRILIUM
INTRODUCTION :
The same relative ease with which we were able to find explicit solution of higher order linear differential
equation with constant coefficient in the preceding sections does not , in general , carry over to linear
equation with variable coefficients . However , the type of differential equations that we consider in this
section is an exception to this rule ; it is a linear equation with variable coefficients whose general solution
can always be expressed in terms of power of x , sines , cosines , and logarithmic functions . Moreover , its
method of solutions is quite similer to that for constant –coefficient equations in that an auxiliary equation
must be solved .

4. Cauchy Euler equation :
A linear differential equation of the form
𝑎𝑛𝑥𝑛 ⅆ𝑛𝑦2 a n −1 𝑥𝑛−1 ⅆⅆ𝑥𝑛𝑛𝑦−Ԧ1 + 𝑎1𝑥
ⅆⅆ𝑦𝑥 + 𝑎0𝑦 = 𝑔 𝑥 ⅆ𝑥
Where the coefficients 𝑎𝑛 , 𝑎𝑛−1……..𝑎0 are constants , is known as a Cauchy Euler equation . The
observable characteristic of this type of equation is that the degree K = n ,n -1 . . . . , 1,0 of the
monomial coefficients 𝑥𝑘 𝑚𝑎𝑡𝑐ℎ𝑒𝑠 𝑡ℎ𝑒 order k of differentiation 𝑑𝑦
𝑘Τⅆ 𝑥𝑘 :
Note the following properties of these equation Any
solution will be on a subset of −∞, 0 or 0, ∞ .
The powers of X must match the order of the

5. derivatives.

6. METHOD OF SOLUTION
We try a solution of the form 𝑦 = 𝑥𝑚 , where m is to be determined . Analogous to what happened
When we substituted ⅇ𝑚𝑥 into a linear equation with constant coefficients substitute 𝑥𝑚 , each term of
a Cauchy – Euler equation becomes a polynominal 𝑥𝑚 , since
𝑎𝑘𝑥𝑘 ⅆ
ⅆ𝑘
𝑋
𝑌𝑘 =𝑎𝑘𝑥𝑘𝑚 𝑚 − 1 …(m-2 )…….(m –k+1 ) 𝑥𝑚−𝑘
𝑎𝑘𝑚 𝑚 − 1 𝑚 − 2 … . . (𝑚 − 𝑘 + 1) 𝑥𝑚
For example , when we substitute y = 𝑥𝑚 , the second order equation becomes

8. FIRST ORDER CAUCHY EULER
Note that
𝑎1 ⅆ𝑥 + 𝑎0𝑦 = 0 ⇒ ⅆ𝑥 = 𝑎01𝑦𝑥 ⇒ ⅆ𝑦𝑦 =
𝑎𝑎10𝑥1 dx ⅆ𝑦 ⅆ𝑦 𝑎
We can separate the variables as seen , and solve for y =𝑚𝑥
EXAMPLE :
We make the following substitution : 𝑥 = ⅇ𝑡 . 𝑇ℎ𝑒𝑛 𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒𝑠 𝑤𝑖𝑙𝑙 𝑏𝑒 .
ⅆ𝑦 ⅆ𝑦
𝑦 = ⅆ𝑦 = ⅆ𝑡 = ⅆ𝑡𝑡 = ⅇ−𝑡
ⅆⅆ𝛾𝑡 ⅆ𝑥 ⅆ𝑥 ⅇ
ⅆ𝑡

10. SECOND ORDER CAUCHY EULER
We now assume we are searching for solutions of the form 𝑦 = 𝑥𝑚 . Sure enough’
𝑑2𝑦 + 𝑏𝑥 ⅆ𝑦
2 + 𝑐𝑦 = 0 ⇒ 𝑥𝑚𝑎𝑚2 +𝑏 − 𝑎𝑚 + 𝑐= 0
𝑎𝑥 2 ⅆ𝑥 𝑑𝑥
Second way of solving an euler equation
In the second method we look for a solution of the equation in the form of the
power function 𝑦 = 𝑥𝑘 where k is an unknown number . It follows from here that
𝑑𝑦 𝑘−1, ⅆ2𝑦2 = 𝑘 (𝑘 −
1)𝑥𝑘−2 = 𝑘𝑥

11. 𝑑𝑥 ⅆ𝑥

13. THIRD ORDER EQUATION
Solve 𝑥3 ⅆ3𝑦3 + 5𝑥2 ⅆⅆ2𝑥𝑦2 + 7x
ⅆⅆ𝑦𝑥 8y=0 ⅆ𝑥
Solution
The first three derivities of y = 𝑥𝑚 are
ⅆ𝑦 𝑚−1 ⅆ2𝑦
= 𝑚𝑥
ⅆ𝑥 ⅆ𝑥2ⅆ𝑥
In this case we see that 𝑦 = 𝑥𝑚 will be a solution of the differential equation for
𝑚1 = 2, 𝑚2 = 2ⅈ anⅆ 𝑚3 = −2ⅈ . Hence the general solution is 𝑦 = 𝑐1𝑥−2 + 𝑐2 cos 2 ln 𝑥 + 𝑐3 sⅈn 21𝑛𝑥

15. CONCLUSION
The development of the solution set of certain ordinary differential equations still remains the object of
Research , with attractive problems and high applicability in the phenomena of nature. It is evident the
difficulty encountered by students to establish a relation of interest with the area of calculation , particularly
In differential equations , perhaps because they do not know the wide field of application that these
equations make available . In the light of the above , it is expected that this work may significantly awaken
other research on Cauchy Euler equation in order to minimize the lags between mathematical abstraction
and its practice .

16. REFERENCES :

17. 1. Boyce , William E ; DiPrima , Richard c .
2. Abhunahman , Sergio Antonio . Equacoes Differenciais . Rio de Janeiro : Livros Tecnicos e Cintificos 1979
3. JR. Wylie , C.R. mathematics superiors para ingenieria . New York : Mcgraw – Hill, 1969
4. BOYER , Carl , B . Historia da mathematica . Sao Paulo : Edgard Blucher 1974

18. THANK YOU

19. ~SUTRIPTA SARKAR