International Research Journal of Engineering and Technology (IRJET)
e-ISSN: 2395-0056
Volume: 11 Issue: 01 | Jan 2024
p-ISSN: 2395-0072
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Solving Linear Differential Equations with Constant Coefficients Prof. Ms. Pooja Pandit Kadam1 , Prof. Swati Appasaheb Patil2, Prof. Akshata Jaydeep Chavan3 Assistant Professor, Dept. Of Sciences & Humanities, Rajarambapu institute of Technology, Maharashtra, India Assistant Professor, Dept. Of Sciences & Humanities, Rajarambapu institute of Technology, Maharashtra, India Assistant Professor, Dept. Of Engineering Sciences , SKN Sinhgad Institute of Technology & Science, Kusgoan India --------------------------------------------------------------------***---------------------------------------------------------------------
Abstract - In this paper, we are discussing about the
different method of finding the solution of the linear differential equations with constant coefficient. For more understanding of different method we have included different examples to find Particular solution of function of linear differential equations with constant coefficient. Particular solution is combination of complementary function and particular integral. Key Words: Particular Integral, Complementary Function, Auxiliary equation.
Otherwise the solution of the F(D)y=X is satisfies the particular integral. The complete solution of the differential equation is Y= C.F. + P.I. i.e. Y=Complementary Function + Particular integral
2. Methods of Finding Complementary Function Consider the equation F(D)y=F(x)
Step 1: Put D=m, Auxiliary equation is given by f(m)=0
1. INTRODUCTION This paper includes overview of the definition and different types of finding solution of the linear differential equations with constant coefficient. Here we mentioned different rules for finding particular solution. We applied the linear differential equation in various science and engineering field. It is applied to electrical circuit, chemical kinetics, Heat transfer, control system, mechanical systems etc. First we see the definitions of linear differential equation with constant coefficient then we see general solution with complementary function and particular integral.
1.1 Linear Differential Equation Definition: A general linear differential equation of nth order with constant coefficients is given by:
Where K ‘s are constant and F(x) is function of x alone or constant.
Complete solution of equation f(D)y=F(x) is given by C.F + P.I. Where C.F. denotes complementary function and P.I. is particular integral. When F(x) =0, then the solution of equation F (D)y=0 is given by y=C.F.
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Table -1: Methods of finding C.F. Methods of finding C.F
Roots of A.E.
Complementary function (C.F.)
If the n roots of A.E. are real and distinct say
C1em1X + C2em2X +…… + Cnemnx
If two or more roots are equal.i.e.
+ …..
.
m1=α+iβ, m2=α-iβ
1.2 Solving Linear Differential Equations with Constant Coefficients:
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Step 3: After we get complementary function according to given rules below:
If roots are imaginary
Or
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Step 2: Solve the auxiliary equation and find the roots of the equation. Lets roots are m1,m2,m3….mn .
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If roots are imaginary and repeated twice m1=m2= α+iβ, m3=m4= α-iβ,
+
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