Communications on Applied Nonlinear Analysis 

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Vol 32 No. 9s (2025) 

  

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Non-Homogeneous Fuzzy Differential Equation with Triangular Fuzzy 

Number as Initial condition 
 

Dnyaneshwar P. Bawane1*, Lemraj S. Ladke2, 
1Department of Applied Mathematics and Humanities, Yeshwantrao Chavan College of Engineering, Nagpur, 

Maharashtra, India. 

E-mail ID: dnyanesh02@gmail.com 

2Department of Mathematics, Nilkanthrao Shinde Science & Arts College, Bhadrawati, Maharashtra, India. 

E-mail ID: lemrajvandana@gmail.com 

 

Article History: 

Received: 12-01-2025 

Revised: 15-02-2025 

Accepted: 01-03-2025 

Abstract:  

A solution of First Order Nonhomogeneous Linear Fuzzy Differential Equation with 

initial condition as Triangular Fuzzy Number is discussed in this paper and to obtained a 

general solution, we have used a method of interval arithmetic on α-cut interval. Here, 

we have proposed a solution Non-Homogeneous Fuzzy Differential Equation by 

considering four different possible cases of real valued functions involved in differential 

equations. At the last, an example of first order nonhomogeneous linear fuzzy 

differential equation under fuzzy initial condition is being solved to verify the result. 

Keywords: Fuzzy Differential Equation, Triangular Fuzzy number, Support and Core of 

Fuzzy set, Interval arithmetic. 

 

1. Introduction 

Fuzzy differential equation is a model of varying situations under vague conditions has an important 

role to various fields of Engineering and Science. The concept of Fuzzy Differential elution  was first 

introduced by Kandel, A. and Byatt, W.J [1] in 1978. Followed by in 1982 Dubois and Prade [2], [3], 

[4] discuss about derivative of crisp function at fuzzy point and derivative of fuzzy function at crisp 

point. In 1985, Kaleva, O. [6] mention a differentiability and integrability of fuzzy valued function 

with exitance and uniqueness its solution. Seikkala, S. [5] has conveyed a use of extension principle 

and extremal solution of initial value problems to solve fuzzy differential equation in 1987. In the 

year 1990, Baidosov, V. [21] and Hullermeier, E [22] has introduced a differential system with fuzzy 

parameter known as fuzzy differential inclusion. In year 1999, Oberguggenberger, M. [23] used 

zadeh’s extension principle get solution to fuzzy differential equation. 

In year 2000, Buckley, et.al. [9] acknowledge a new solution of first order fuzzy initial value 

problem depends on various categories of derivatives of fuzzy function. Park, J. Han, H. [10] 

discussed a unique fuzzy solution of differential equation under fuzzy circumference applying 

successive approximation method . In 2008, Chaco-Cano, Y. and Romain-Flores, H. [11], [24] 

Proposed a new solution to the Fuzzy differential equation based in generalised H-differentiability 

and π- derivative of function. In 2010, Bede. B. et.al [12], [13] used L-R type fuzzy number as 

Starting situation of fuzzy differential equation and also proposed two general solutions for fuzzy 

differential equation with existence. Duraisamy, C. and Usha,B. [14] solved a fuzzy initial value 

problem of first order by using third order Runge-Kutta method with Trapezoidal fuzzy number as 

initial condition. Salahshour, S. [15] In 2011, has promotes uniqueness and existence of solution of 

Nth-order fuzzy differential equation generalised differentiability in Banach space. Plotnikov, A. 

mailto:dnyanesh02@gmail.com
mailto:lemrajvandana@gmail.com


Communications on Applied Nonlinear Analysis 

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with Skripnik, N. [25] In 2012, emphasized on new concept of fuzzy function generalized derivative 

and existence of a fuzzy differential equation. In year 2013, Mondal, S. et.al [16], [18] has discussed 

a solution of homogeneous first order linear fuzzy differential equation using method of Lagrange 

multiplier and intuitionistic Triangular fuzzy number and in year 2014, they [27] have promotes 

solution of Nonhomogeneous Fuzzy differential equation using Lagrange multiplier method. In year 

2020 Alamin, A.  et.al. [26] analyse a Solution of non-homogeneous difference equation and in 2023, 

Padmapriya, V., & Kaliyappan, M. [28] has discussed a solution of system of non-homogeneous 

fuzzy fractional differential equations. In 2024, Bawane D.P. and Ladke L.S. [29] has discus a 

solution for the linear fuzzy differential equation of first order using interval arithmetic techniques.  

2. Preliminaries  

Definition 2.1: Fuzzy Set 

A set 𝐴 define on universe of discourse 𝑋 is said to be fuzzy if ∀ 𝑡 ∈ 𝑋 , membership grade of 

belongings is lies between [0,1] ⊂ ℝ  and it is denoted and define as  

𝐴 = {
𝜇𝐴 (𝑡)

𝑡
| 𝑡 ∈ 𝑋 , 𝜇𝐴 (𝑡) ∈ [0,1] }. Where, 𝜇𝐴: 𝑋 → [0,1] is the membership grade. 

 

Definition 2.2: α-cut of Fuzzy set 

A crisp set obtained from a fuzzy set which containing all elements whose membership grade is 

higher than or equal to the state value 𝛼 ∈ [0,1] is known as α-cut of Fuzzy set. It is represented as as 

  A𝛼 = {𝑡𝑘|𝑡𝑘 ∈ 𝑋 𝑎𝑛𝑑 𝜇𝐴(𝑡𝑘) ≥ 𝛼}. 
 

Definition 2.3: Fuzzy Set Support 

A crisp set obtained from a fuzzy set with all members whose membership grade is more than 0 is 

known Fuzzy set support. It is represented as,  Supp(𝐴) = {𝑡   |  𝑡 ∈ 𝑋 𝑎𝑛𝑑 𝜇𝐴(𝑡) > 0}. 
 

Definition 2.4: Fuzzy Set Core 

It is Classical Set of elements 𝑡 ∈ 𝑋 whose membership grade of belongingness in fuzzy set 𝐴 equal 

to 1 and it is denoted as follows 

 Core(𝐴) = {𝑡   | 𝑡 ∈ 𝑋 𝑎𝑛𝑑 𝜇𝐴(𝑡) = 1}. 
 

Definition 2.5: Arithmetic’s on closed Intervals 

Consider 𝐴 = [𝑎1, 𝑎2]  and 𝐵 = [𝑏1, 𝑏2] are two closed intervals explain on real number set ℝ in 

such a way that 𝑎1 ≤ 𝑎2 and 𝑏1 ≤ 𝑏2, the arithmetic’s such as subtraction  and addition are as 

follows.  

a) For Subtraction: 𝐴 − 𝐵 = [𝑎1, 𝑎2] − [𝑏1, 𝑏2] = [(𝑎1 − 𝑏2)  , (𝑎2 − 𝑏1)]. 

b) For Addition: 𝐴 + 𝐵 = [𝑎1, 𝑎2] + [𝑏1, 𝑏2] = [(𝑎1 + 𝑏1)  , (𝑎2 + 𝑏2)] 
 

Definition 2.6: Fuzzy Number 

A fuzzy set 𝐴 defines on set of real number in ℝ is said to be a fuzzy number as long as it fulfil the 

circumstances as below 

(i) Set 𝐴 is normalize fuzzy set i.e. core of fuzzy set is non-void 

(ii) Support of fuzzy set 𝐴 is constrained.   

      i.e. for real numbers  𝜆1, 𝜆2 ∈  ℝ  there exist  𝑡 ∈ (−∞, 𝜆1) & (𝜆2 , ∞) such that  𝜇𝐴(t) = 0. 

(iii) Every α-cut set are closed intervals. 



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      i.e. for every  ∈ [0,1], 𝐴α = [𝑎α,   𝑏α] is closed interval. 

 

Definition 2.7: Triangular Fuzzy Number (TNF) 

A Triangular Fuzzy Number ‘A’ characterize with three points [𝑎1  ,  𝑎2,  𝑎3] with membership 

function 

 𝜇𝐴(𝑡) =

{
 
 

 
 
0,           𝑖𝑓 𝑡 < 𝑎1  𝑎𝑛𝑑  𝑡 >  𝑎3
𝑡 − 𝑎1  
𝑎2 − 𝑎1  

 ,          𝑖𝑓    𝑎1  ≤ 𝑡 ≤  𝑎2

 𝑎3 − 𝑡

 𝑎3 − 𝑎2  
,          𝑖𝑓    𝑎2  ≤ 𝑡 ≤  𝑎3

 

and α-cut set  is 𝐴𝛼 = [𝑎1  + 𝛼( 𝑎2 −  𝑎1 ),    𝑎3  − 𝛼( 𝑎3 −  𝑎2 )] = [𝑎1  + αℒ,   𝑎3  − αℛ],  
here ℒ = ( 𝑎2 −  𝑎1 ) > 0 & ℛ = ( 𝑎3 −  𝑎2 ) > 0. 

If ℒ = ℛ , triangular fuzzy number A = [𝑎1  ,  𝑎2,  𝑎3] is symmetric in nature. 

 

Definition 2.8: First Order Homogeneous Fuzzy Differential Equation 

Equation (𝑡, 𝑦,
𝑑𝑦

𝑑𝑡
 ) = 0 , 𝑦(𝑡0) = 𝑦0 is homogeneous fuzzy differential equation of first order, if 

initial value 𝑦0 is fuzzy valued  number and hence its  solution is also a fuzzy solution.  

Definition 2.9: Nonhomogeneous First Order Fuzzy Differential Equation  

Equation (𝑡, 𝑦,
𝑑𝑦

𝑑𝑡
 ) ≠ 0 , 𝑦(𝑡0) = 𝑦0 is nonhomogeneous fuzzy differential equation of first order, 

where  initial value 𝑦0 is fuzzy valued number and hence its  solution is also a fuzzy solution. 

Definition 2.10: Strong and Weak Solution  

Let  𝑓 (𝑡, 𝑦,
𝑑𝑦

𝑑𝑡
 ) = 0 ≠ 0 , 𝑦(𝑡0) = 𝑦0 is the fuzzy differential equation of first order, where initial 

condition   𝑦0 is fuzzy value  then solution 𝑦(𝑡) is strong solution provided  it’s α-cut, 

 𝑦(𝑡, α) = [𝑦1(𝑡, α) , 𝑦2(𝑡, α)] satisfy following conditions (I) 
𝜕𝑦1(𝑡,α)

𝜕α
> 0 and (II)  

𝜕𝑦2(𝑡,α)

𝜕α
< 0, 

otherwise it is a weak solution. 

 

3. Case Study 

Consider a Non-Homogeneous First Order Linear Fuzzy Differential Equation  

𝑑𝑦(𝑡)

𝑑𝑡
+ 𝑃(𝑡)𝑦(𝑡) = 𝑄(𝑡)                                                                                                                     (1) 

with 𝑦 ̃(𝑡) = 𝑦(𝑡0) = [𝑎1,  𝑎2  , 𝑎3] Is TFN.  Here 𝑃(𝑡) and 𝑄(𝑡) are any functions of real valued 

variable 𝑡. The solution of equation (1) is value of dependent variable 𝑦(𝑡) whose α-cut interval is 

denoted as  𝑦(𝑡) = [ 𝑦1(𝑡, 𝛼), 𝑦2(𝑡, 𝛼) ] and the α-cut interval of given initial condition is denoted 

as  𝑦(𝑡0) = [𝑎1  + αℒ,   𝑎3  − αℛ] = [ 𝑦1(𝑡0, 𝛼), 𝑦2(𝑡0, 𝛼) ]  

 where, ℒ = ( 𝑎2 −  𝑎1 ) > 0 and ℛ = ( 𝑎3 −  𝑎2 ) > 0.  

Case-I: If 𝑃(𝑡)  and  𝑄(𝑡)are any real valued functions 

An α-cut of differential equation (1) is 



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[
𝑑𝑦1(𝑡,∝)

𝑑𝑡
,
𝑑𝑦2(𝑡,∝)

𝑑𝑡
 ] + 𝑃(𝑡)[𝑦1(𝑡, ∝), 𝑦2(𝑡, ∝)] = 𝑄(𝑡)  

i.e.    
𝑑𝑦1(𝑡,∝)

𝑑𝑡
+ 𝑃(𝑡)𝑦1(𝑡, ∝) = 𝑄(𝑡)            (2) 

         
𝑑𝑦2(𝑡,∝)

𝑑𝑡
+ 𝑃(𝑡)𝑦2(𝑡, ∝) = 𝑄(𝑡)                (3) 

Since 𝑃(𝑡) is real valued function, the integrating factor is  𝐼(𝑡) = 𝑒∫𝑃(𝑡)𝑑𝑡  And solutions are  

𝑦1(𝑡, ∝)  𝐼(𝑡) = ∫𝑄(𝑡)  𝐼(𝑡)𝑑𝑡  + 𝐶1                                                                                     (4) 

𝑦2(𝑡, ∝)  𝐼(𝑡) = ∫𝑄(𝑡)  𝐼(𝑡)𝑑𝑡  + 𝐶2                                                                                      (5) 

at 𝑡 = 𝑡0,    𝐶1 = 𝑦1(𝑡0, ∝)  𝐼(𝑡0) − (∫𝑄(𝑡)  𝐼(𝑡)𝑑𝑡 )𝑡=𝑡0   and 

 𝐶2 = 𝑦2(𝑡0, ∝)  𝐼(𝑡0) − (∫𝑄(𝑡)  𝐼(𝑡)𝑑𝑡 )𝑡=𝑡0    

and from equation (4) and (5) we get 

𝑦1(𝑡, ∝)   = (𝐼(𝑡))
−1
{∫𝑄(𝑡)  𝐼(𝑡)𝑑𝑡  + (𝑎1  + αℒ)  𝐼(𝑡0) − (∫ 𝑄(𝑡)  𝐼(𝑡)𝑑𝑡 )𝑡=𝑡0}                      (6) 

𝑦2(𝑡, ∝)   = (𝐼(𝑡))
−1
{∫𝑄(𝑡)  𝐼(𝑡)𝑑𝑡  + (𝑎3  − αℛ)  𝐼(𝑡0) − (∫𝑄(𝑡)  𝐼(𝑡)𝑑𝑡 )𝑡=𝑡0}                      (7) 

are satisfying a condition of strong solution i.e.  
𝜕𝑦1(𝑡,α)

𝜕α
= ℒ𝐼(𝑡0)(𝐼(𝑡))

−1
> 0  

and   
𝜕𝑦2(𝑡,α)

𝜕α
= −ℛ𝐼(𝑡0)(𝐼(𝑡))

−1
< 0. 

Hence, based on α-cut interval arithmetic, the solution of differential equation (1) as follows  𝑦(𝑡) =
 [𝑦1(𝑡, ∝) ,     𝑦1(𝑡, ∝),     𝑦2(𝑡, ∝), 𝑦2(𝑡, ∝)] 

         = (𝐼(𝑡))
−1
{∫ 𝑄(𝑡)  𝐼(𝑡)𝑑𝑡  + 𝑦(𝑡0) 𝐼(𝑡0) − (∫𝑄(𝑡)  𝐼(𝑡)𝑑𝑡 )𝑡=𝑡0}                                          (8) 

Case-II: If 𝑃(𝑡) is real valued function and  𝑄(𝑡) is fuzzy function 

Let 𝑄(𝑡) is fuzzy valued function such that  

 𝑖𝑛𝑓. 𝑄(𝑡) = 𝑄1(𝑡) and  𝑠𝑢𝑝. 𝑄(𝑡) = 𝑄2(𝑡) 

Then,  α-cut of differential equation (1) is 

[
𝑑𝑦1(𝑡,∝)

𝑑𝑡
,
𝑑𝑦2(𝑡,∝)

𝑑𝑡
 ] + 𝑃(𝑡)[𝑦1(𝑡, ∝), 𝑦2(𝑡, ∝)] = [𝑄1(𝑡, ∝), 𝑄2(𝑡, ∝)]  

i.e.    
𝑑𝑦1(𝑡,∝)

𝑑𝑡
+ 𝑃(𝑡)𝑦1(𝑡, ∝) = 𝑄1(𝑡, ∝)                           (9) 

         
𝑑𝑦2(𝑡,∝)

𝑑𝑡
+ 𝑃(𝑡)𝑦2(𝑡, ∝) = 𝑄2(𝑡, ∝)                       (10) 

Since 𝑃(𝑡) is real valued function, the integrating factor is  𝐼(𝑡) = 𝑒∫𝑃(𝑡)𝑑𝑡  And solutions are  

𝑦1(𝑡, ∝)  𝐼(𝑡) = ∫𝑄1(𝑡, ∝)  𝐼(𝑡)𝑑𝑡  + 𝐶1                                                                                          (11) 

𝑦2(𝑡, ∝)  𝐼(𝑡) = ∫𝑄2(𝑡, ∝)   𝐼(𝑡)𝑑𝑡  + 𝐶2                                                                                         (12) 

at 𝑡 = 𝑡0,    𝐶1 = 𝑦1(𝑡0, ∝)  𝐼(𝑡0) − (∫𝑄1(𝑡, ∝)  𝐼(𝑡)𝑑𝑡 )𝑡=𝑡0   and 



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 𝐶2 = 𝑦2(𝑡0, ∝)  𝐼(𝑡0) − (∫𝑄2(𝑡, ∝)  𝐼(𝑡)𝑑𝑡 )𝑡=𝑡0    

and from equation (11) and (12) we get 

𝑦1(𝑡, ∝)   = (𝐼(𝑡))
−1
{∫𝑄1(𝑡, ∝)  𝐼(𝑡)𝑑𝑡  + (𝑎1  + αℒ)  𝐼(𝑡0) − (∫𝑄1(𝑡, ∝)  𝐼(𝑡)𝑑𝑡 )𝑡=𝑡0}           (13) 

𝑦2(𝑡, ∝)   = (𝐼(𝑡))
−1
{∫𝑄2(𝑡, ∝)  𝐼(𝑡)𝑑𝑡  + (𝑎3  − αℛ)  𝐼(𝑡0) − (∫𝑄2(𝑡, ∝)  𝐼(𝑡)𝑑𝑡 )𝑡=𝑡0}          (14) 

are satisfying a condition of strong solution i.e.  
𝜕𝑦1(𝑡,α)

𝜕α
> 0 and   

𝜕𝑦2(𝑡,α)

𝜕α
< 0. 

Hence, based on α-cut interval arithmetic, the solution of differential equation (1) as follows  𝑦(𝑡) =
 [𝑦1(𝑡, ∝) ,     𝑦1(𝑡, ∝),     𝑦2(𝑡, ∝), 𝑦2(𝑡, ∝)] 

         = (𝐼(𝑡))
−1
{∫[𝑄1(𝑡),𝑄2(𝑡) ]  𝐼(𝑡)𝑑𝑡  + 𝑦(𝑡0) 𝐼(𝑡0) − (∫[𝑄1(𝑡), 𝑄2(𝑡) ]   𝐼(𝑡)𝑑𝑡 )𝑡=𝑡0}        (15) 

Where,     𝑄1(𝑡) = 𝑀𝑖𝑛{𝑄1(𝑡, 0), 𝑄1(𝑡, 1), 𝑄2(𝑡, 0), 𝑄2(𝑡, 1)} and     

                   𝑄2(𝑡) = 𝑀𝑎𝑥{𝑄1(𝑡, 0), 𝑄1(𝑡, 1), 𝑄2(𝑡, 0), 𝑄2(𝑡, 1)} 

Case-III: If 𝑃(𝑡) is fuzzy function and  𝑄(𝑡) is real function 

Let 𝑃(𝑡) is fuzzy valued function such that  𝑖𝑛𝑓. 𝑃(𝑡) = 𝑃1(𝑡) and  𝑠𝑢𝑝. 𝑃(𝑡) = 𝑃2(𝑡) 

Then,  α-cut of differential equation (1) is 

[
𝑑𝑦1(𝑡,∝)

𝑑𝑡
,
𝑑𝑦2(𝑡,∝)

𝑑𝑡
 ] + [𝑃1(𝑡, ∝), 𝑃2(𝑡, ∝)] [𝑦1(𝑡, ∝), 𝑦2(𝑡, ∝)] = 𝑄(𝑡)  

i.e.    
𝑑𝑦1(𝑡,∝)

𝑑𝑡
+ 𝑃1(𝑡, ∝)𝑦1(𝑡, ∝) = 𝑄(𝑡)                                           (16) 

         
𝑑𝑦2(𝑡,∝)

𝑑𝑡
+ 𝑃2(𝑡, ∝)𝑦2(𝑡, ∝) = 𝑄(𝑡)                                                     (17) 

Since 𝑃(𝑡) is fuzzy valued function, the integrating factors are  𝐼1(𝑡, ∝) = 𝑒
∫𝑃1(𝑡,∝)𝑑𝑡  and 

 𝐼2(𝑡, ∝) = 𝑒∫𝑃2(𝑡,∝)𝑑𝑡  and hence solutions are  

𝑦1(𝑡, ∝)  𝐼1(𝑡, ∝) = ∫𝑄(𝑡)  𝐼1(𝑡, ∝)𝑑𝑡  + 𝐶1                                                                                    (18) 

𝑦2(𝑡, ∝)  𝐼2(𝑡, ∝) = ∫𝑄(𝑡)  𝐼2(𝑡, ∝)𝑑𝑡  + 𝐶2                                                                                               

(19) 

at 𝑡 = 𝑡0,    𝐶1 = 𝑦1(𝑡0, ∝)   𝐼1(𝑡0, ∝) − (∫𝑄(𝑡)  𝐼1(𝑡, ∝)𝑑𝑡 )𝑡=𝑡0   and 

 𝐶2 = 𝑦2(𝑡0, ∝) 𝐼2(𝑡0, ∝) − (∫𝑄(𝑡)  𝐼2(𝑡, ∝)𝑑𝑡 )𝑡=𝑡0    

and from equation (18) and (19) we get 

𝑦1(𝑡, ∝)   = (𝐼1(𝑡, ∝))
−1
{∫𝑄(𝑡)  𝐼1(𝑡, ∝)𝑑𝑡  + (𝑎1  + αℒ)  𝐼1(𝑡0 , ∝) − (∫𝑄(𝑡)   𝐼1(𝑡, ∝)𝑑𝑡 )𝑡=𝑡0}   

                                                                                                                                                           (20) 

𝑦2(𝑡, ∝)   = (𝐼2(𝑡, ∝))
−1
{∫𝑄(𝑡) 𝐼2(𝑡, ∝)𝑑𝑡  + (𝑎3  − αℛ) 𝐼2(𝑡0 , ∝) − (∫𝑄(𝑡)  𝐼2(𝑡, ∝)𝑑𝑡 )𝑡=𝑡0}      

                                                                              (21) 



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are satisfying a condition of strong solution i.e.  
𝜕𝑦1(𝑡,α)

𝜕α
> 0 and   

𝜕𝑦2(𝑡,α)

𝜕α
< 0. 

Hence, based on α-cut interval arithmetic, the solution of differential equation (1) as follows  𝑦(𝑡) =
 [𝑦1(𝑡, ∝) ,     𝑦1(𝑡, ∝),     𝑦2(𝑡, ∝), 𝑦2(𝑡, ∝)] 

= (𝐼(𝑡, ∝))
−1
{∫𝑄(𝑡)  𝐼(𝑡, ∝)𝑑𝑡  + 𝑦(𝑡0) 𝐼(𝑡0, ∝ ) − (∫𝑄(𝑡) 𝐼(𝑡, ∝)𝑑𝑡 )𝑡=𝑡0}                               (22) 

Where,  (𝐼(𝑡, ∝)) = [(𝐼1(𝑡, ∝)), (𝐼2(𝑡, ∝))]                  

Here,  𝐼1(𝑡) = 𝑀𝑖𝑛{𝐼1(𝑡, 0), 𝐼1(𝑡, 1), 𝐼2(𝑡, 0), 𝐼2(𝑡, 1)} and  

            𝐼2(𝑡) = 𝑀𝑎𝑥{𝐼1(𝑡, 0), 𝐼1(𝑡, 1), 𝐼2(𝑡, 0), 𝐼2(𝑡, 1)} 

Case-IV: If 𝑃(𝑡) and  𝑄(𝑡) are fuzzy function  

Let 𝑄(𝑡) is fuzzy valued function such that  𝑖𝑛𝑓. 𝑄(𝑡) = 𝑄1(𝑡) and  𝑠𝑢𝑝. 𝑄(𝑡) = 𝑄2(𝑡) 

Let 𝑃(𝑡) is fuzzy valued function such that  𝑖𝑛𝑓. 𝑃(𝑡) = 𝑃1(𝑡) and  𝑠𝑢𝑝. 𝑃(𝑡) = 𝑃2(𝑡) 

Then,  α-cut of differential equation (1) is 

[
𝑑𝑦1(𝑡,∝)

𝑑𝑡
,
𝑑𝑦2(𝑡,∝)

𝑑𝑡
 ] + [𝑃1(𝑡, ∝), 𝑃2(𝑡, ∝)] [𝑦1(𝑡, ∝), 𝑦2(𝑡, ∝)] = [𝑄1(𝑡, ∝), 𝑄2(𝑡, ∝)]  

i.e.    
𝑑𝑦1(𝑡,∝)

𝑑𝑡
+ 𝑃1(𝑡, ∝)𝑦1(𝑡, ∝) = 𝑄1(𝑡, ∝)                                          (23) 

         
𝑑𝑦2(𝑡,∝)

𝑑𝑡
+ 𝑃2(𝑡, ∝)𝑦2(𝑡, ∝) = 𝑄2(𝑡, ∝)                                         (24) 

Since 𝑃(𝑡) is fuzzy valued function, the integrating factors are  𝐼1(𝑡, ∝) = 𝑒
∫𝑃1(𝑡,∝)𝑑𝑡  and 

 𝐼2(𝑡, ∝) = 𝑒∫𝑃2(𝑡,∝)𝑑𝑡  and hence solutions are  

𝑦1(𝑡, ∝)  𝐼1(𝑡, ∝) = ∫𝑄1(𝑡, ∝)  𝐼1(𝑡, ∝)𝑑𝑡  + 𝐶1                                                                               (25) 

𝑦2(𝑡, ∝)  𝐼2(𝑡, ∝) = ∫𝑄2(𝑡, ∝)  𝐼2(𝑡, ∝)𝑑𝑡  + 𝐶2                                                                              (26) 

at 𝑡 = 𝑡0,    𝐶1 = 𝑦1(𝑡0, ∝)   𝐼1(𝑡0, ∝) − (∫𝑄1(𝑡, ∝) 𝐼1(𝑡, ∝)𝑑𝑡 )𝑡=𝑡0   and 

 𝐶2 = 𝑦2(𝑡0, ∝) 𝐼2(𝑡0, ∝) − (∫𝑄2(𝑡, ∝)  𝐼2(𝑡, ∝)𝑑𝑡 )𝑡=𝑡0    

and from equation (18) and (19) we get 

𝑦1(𝑡, ∝)   = (𝐼1(𝑡, ∝))
−1
{∫𝑄1(𝑡, ∝)  𝐼1(𝑡, ∝)𝑑𝑡  + (𝑎1  + αℒ)  𝐼1(𝑡0 , ∝) − (∫𝑄1(𝑡, ∝

)   𝐼1(𝑡, ∝)𝑑𝑡 )𝑡=𝑡0}       (27) 

𝑦2(𝑡, ∝)   = (𝐼2(𝑡, ∝))
−1
{∫𝑄2(𝑡, ∝) 𝐼2(𝑡, ∝)𝑑𝑡  + (𝑎3  − αℛ) 𝐼2(𝑡0 , ∝) − (∫𝑄2(𝑡, ∝

) 𝐼2(𝑡, ∝)𝑑𝑡 )𝑡=𝑡0}         (28) 

are satisfying a condition of strong solution i.e.  
𝜕𝑦1(𝑡,α)

𝜕α
> 0 and   

𝜕𝑦2(𝑡,α)

𝜕α
< 0. 

Hence, based on α-cut interval arithmetic, a solution of differential equation (1) is as follows  𝑦(𝑡) =
 [𝑦1(𝑡, ∝) ,     𝑦1(𝑡, ∝),     𝑦2(𝑡, ∝), 𝑦2(𝑡, ∝)] 

= (𝐼(𝑡, ∝))
−1
{∫𝑄(𝑡)  𝐼(𝑡, ∝)𝑑𝑡  + 𝑦(𝑡0) 𝐼(𝑡0, ∝ ) − (∫𝑄(𝑡) 𝐼(𝑡, ∝)𝑑𝑡 )𝑡=𝑡0}                              (29) 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 9s (2025) 

  

2468 
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here,  (𝐼(𝑡, ∝)) = [(𝐼1(𝑡, ∝)), (𝐼2(𝑡, ∝))]       𝑄(𝑡) = [𝑄1(𝑡),𝑄2(𝑡) ]           

Here,  𝐼1(𝑡) = 𝑀𝑖𝑛{𝐼1(𝑡, 0), 𝐼1(𝑡, 1), 𝐼2(𝑡, 0), 𝐼2(𝑡, 1)},  

            𝐼2(𝑡) = 𝑀𝑎𝑥{𝐼1(𝑡, 0), 𝐼1(𝑡, 1), 𝐼2(𝑡, 0), 𝐼2(𝑡, 1)}, 

           𝑄1(𝑡) = 𝑀𝑖𝑛{𝑄1(𝑡, 0), 𝑄1(𝑡, 1), 𝑄2(𝑡, 0), 𝑄2(𝑡, 1)} and     

          𝑄2(𝑡) = 𝑀𝑎𝑥{𝑄1(𝑡, 0), 𝑄1(𝑡, 1), 𝑄2(𝑡, 0), 𝑄2(𝑡, 1)}. 

Solved Example: 

Ex.: Consider non homogeneous differential equation  
𝑑𝑦(𝑡)

𝑑𝑡
+ 2𝑦(𝑡) = 𝑡  with starting condition  

        𝑦0 = 𝑦(𝑡0) = 𝑦(0) = [0,1,2] is TFN.  

Here 𝑃(𝑡) = 2 and 𝑄(𝑡) = 𝑡 are real valued functions  

Then, an integrating factor is 𝐼(𝑡) = 𝑒∫2𝑑𝑡 = 𝑒2𝑡 and at 𝑡 = 𝑡0 = 0, 𝐼(𝑡0) = 𝑒0 = 1 

and hence its solution from (8) is 

𝑦(𝑡) =  (𝐼(𝑡))
−1
{∫𝑄(𝑡)  𝐼(𝑡)𝑑𝑡  + 𝑦(𝑡0) 𝐼(𝑡0) − (∫𝑄(𝑡)  𝐼(𝑡)𝑑𝑡 )𝑡=𝑡0}                 

         = 𝑒−2𝑡{∫ 𝑡  𝑒2𝑡𝑑𝑡  + [0,1,2]  − (∫ 𝑡  𝑒2𝑡𝑑𝑡 )𝑡=0}                          

        = 𝑒−2𝑡 {(
𝑡 

2
−
1

4
) 𝑒2𝑡  + [0,1,2]  − ((

𝑡 

2
−

1

4
) 𝑒2𝑡 )

𝑡=0
}       

       = 𝑒−2𝑡 {(
𝑡 

2
−
1

4
) 𝑒2𝑡  + [0,1,2] +

1

4
}         

       = 𝑒−2𝑡 {(
𝑡 

2
−
1

4
) 𝑒2𝑡  + [

1

4
,
5

4
,
9

4
]}  is the required solution                          

4. Conclusion 

This paper promotes a solution of Nonhomogeneous First Order Linear Fuzzy Differential Equation 

with starting value condition as Triangular Fuzzy Number. Here, we have considered four different 

possible cases of functions involved in differential equations. In this paper we solve a random 

example with triangular fuzzy number as initial condition and obtained a strong solution by case-I. 

Here we propose that for other cases of functions involved in differential equations, the method is 

appropriate and may get strong solution.  Further we propose to solve first order Nonhomogeneous 

Differential Equation under different fuzzy numbers as Initial value 

5. Acknowledgement 

We are extremely thankful and oblige to the reviewers of this paper and the editors of journal for 

their valuable remarks which encourages us to put some ideas in front of the world and help us to 

advance the demonstration. 

 

 

 



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