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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.
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