Kokanduni uz

Bernoulli's equation
Definition: An equation of this forms
𝑦
′
+ 𝑃(𝑥)𝑦 = 𝑄(𝑥)𝑦
𝛼
(𝛼 ≠ 0; 𝛼 ≠ 1)
(6) 
is called Bernoulli's equation. Bernoulli's equation using substitution 
𝑦 = 𝑢(𝑥) ⋅ 𝑣(𝑥)
or 
Lagrange's method can be solved directly with Also the substitution 
𝑧 = 𝑦
1−𝛼
using equation 
(6) is reduced to a linear differential equation.
Example 2.
Solve the equation 
𝑥𝑦
′
− 4𝑢 = 𝑥
2
√𝑦
. After dividing the equation by 
𝑋
, we 
get the equation 
𝑦
′
− 4
𝑦
𝑥
= 𝑥
√
𝑦 .
We look for the solution in the form 
𝑦 = 𝑢(𝑥) ⋅ 𝑣(𝑥)
. Then, 
coming to the equation 
𝑢
′
𝑣 + 𝑢
(
𝑣
′
−
4
𝑥
𝑣
)
= 𝑥
√
𝑢𝑣
,

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𝑣
′
−
4
𝑥
𝑣 = 0,
𝑑𝑣
𝑣
=
4𝑑𝑥
𝑥
,
𝑙𝑛 |𝑣| = 4𝑙𝑛 |𝑥|, 𝑣 = 𝑥
4
;
𝑥
4
𝑢
′
= 𝑥
3
√
𝑢, 𝑢
−
1
2
𝑑𝑢 =
𝑑𝑥
𝑥
;
2𝑢
1
2
= 𝑙𝑛 |𝑥| + 𝐶, 𝑢 =
(𝑙𝑛 |𝑥| + 𝐶)
2
4
;
𝑦 =
𝑥
4
(𝑙𝑛 |𝑥| + 𝐶)
2
4
can be solved. 
Conclusion: 
In this paper, we have provided an overview of first-order linear 
differential equations and Bernoulli's differential equation. We have shown how to solve these 
equations using integrating factors and transformations, respectively. Additionally, we have 
reviewed some of the literature on these topics, including works by Simmons and Krantz, 
Pelesko and Bernstein, and Edwards. Differential equations are a fundamental tool in many 
fields of science and engineering and the study of first-order. 
References: 
1.
Edwards, C. H. (2010). Differential equations and linear algebra. Pearson. 
2.
Pelesko, J. A., & Bernstein, D. H. (1998). Bernoulli's differential equation revisited. 
Mathematical and Computer Modelling, 27(6), 49-59. 
3.
Simmons, G. F., & Krantz, S. G. (2006). Differential equations with applications and 
historical notes. McGraw-Hill. 
4.
Stewart, J. (2015). Calculus: Early transcendentals. Cengage Learning. 
5.
Tenenbaum, M., & Pollard, H. (1985). Ordinary differential equations. Dover 
Publications.