Case
Ejercicios de ecuaciones reversibles y homogéneas
- Course
- Institution
Ejercicios de ecuaciones reversibles y homogéneas
[Show more]
Seller
Follow
jocelynmarcial30
Content preview
Método de reducibles a variables separables.Las ecuaciones diferenciales reducibles a variables separables son en esencia el mismo método sólo que
aplicando previamente un cambio de variable.
Ejemplo 13.
Resuelva la ecuación diferencial reducible a variables separables siguiente
(𝑥 + 𝑦 + 1)𝑑𝑥 = (2𝑥 + 2𝑦 − 1)𝑑𝑦
Solución.
(𝑥 + 𝑦 + 1)𝑑𝑥 = (2𝑥 + 2𝑦 − 1)𝑑𝑦 si 𝑢 = 𝑥 + 𝑦 entonces y
𝑑 𝑑𝑢 𝑑𝑦 𝑑𝑦 𝑑𝑢
(𝑢 = 𝑥 + 𝑦); = 1+ , = −1
𝑑𝑥 𝑑𝑥 𝑑𝑥 𝑑𝑥 𝑑𝑥
(2𝑥 + 2𝑦 − 1)𝑑𝑦 = (𝑥 + 𝑦 + 1)𝑑𝑥
𝑑𝑦 𝑥+𝑦+1 𝑑𝑢 𝑢+1
= sustituyendo queda como −1 =
𝑑𝑥 2𝑥+2𝑦−1 𝑑𝑥 2𝑢−1
𝑑𝑢 𝑢+1 𝑑𝑢 𝑢+1+2𝑢−1 𝑑𝑢 3𝑢
= +1; = ; = separando las variables queda
𝑑𝑥 2𝑢−1 𝑑𝑥 2𝑢−1 𝑑𝑥 2𝑢−1
2𝑢−1
𝑑𝑢 = 𝑑𝑥 aplicando el operador de integración en ambos lados de la ecuación diferencial
3𝑢
2𝑢−1
∫ 𝑑𝑢 = ∫ 𝑑𝑥 ; dividiendo el lado izquierdo de la igualdad
3𝑢
2 1 2 1 1
∫ (3 − 3𝑢) 𝑑𝑢 = ∫ 𝑑𝑥 + 𝐶 ; 3
∫ 𝑑𝑢 − 3 ∫ 𝑢 𝑑𝑢 = ∫ 𝑑𝑥 + 𝐶
2 1
𝑢 − ln(𝑢) = 𝑥 + 𝐶; 2𝑥 + 2𝑦 − ln(𝑥 + 𝑦) = 𝑥 + 𝐶
3 3
2𝑦 − 𝑥 − ln(𝑥 + 𝑦) = 𝐶
Ejemplo 14.
Halle la solución de la siguiente ecuación diferencial por reducibles a variables separables
(𝑥 2 𝑦 3 + 𝑦 + 𝑥 − 2)𝑑𝑥 + (𝑥 3 𝑦 2 + 𝑥)𝑑𝑦 = 0 considere que 𝑢 = 𝑥𝑦
Solución.
(𝑥 2 𝑦 3 + 𝑦 + 𝑥 − 2)𝑑𝑥 + (𝑥 3 𝑦 2 + 𝑥)𝑑𝑦 = 0
𝑑𝑦 −(𝑥 2 𝑦 3 +𝑦+𝑥−2)
(𝑥 3 𝑦 2 + 𝑥)𝑑𝑦 = −(𝑥 2 𝑦 3 + 𝑦 + 𝑥 − 2)𝑑𝑥; =
𝑑𝑥 𝑥 3 𝑦 2 +𝑥
𝑑𝑢
𝑢 𝑑𝑦 𝑥 −𝑢
𝑦= ; = 𝑑𝑥
igualando las derivadas y sustituyendo el valor de 𝑦.
𝑥 𝑑𝑥 𝑥2
𝑑𝑢 𝑢 𝑢 𝑑𝑢 𝑢3 𝑢 𝑑𝑢 𝑢3 +𝑢+𝑥2 −2𝑥
𝑥 −𝑢 −(𝑥 2 ( )3 + +𝑥−2) 𝑥 −𝑢 −( + +𝑥−2) 𝑥 −𝑢 −( )
𝑑𝑥
= 𝑥
𝑢
𝑥
; 𝑑𝑥
= 𝑥 𝑥 𝑑𝑥
; = 𝑥
𝑥2 𝑥 3 ( )2 +𝑥 𝑥2 𝑥𝑢2 +𝑥 𝑥2 𝑥𝑢2 +𝑥
𝑥
𝑑𝑢
𝑥 −𝑢 −(𝑢3 +𝑢+𝑥 2 −2𝑥) 𝑑𝑢 −𝑢3 −𝑢−𝑥 2 +2𝑥
𝑑𝑥
= ; 𝑥 = + 𝑢;
𝑥2 𝑥 2 (𝑢2 +1) 𝑑𝑥 𝑢2 +1
𝑑𝑢 −𝑢3 −𝑢−𝑥 2 +2𝑥+𝑢3 +𝑢 𝑑𝑢 −𝑥 2 +2𝑥
𝑥 = ; 𝑥 = ; (𝑢2 + 1)𝑑𝑢 = (−𝑥 + 2)𝑑𝑥
𝑑𝑥 𝑢2 +1 𝑑𝑥 𝑢2 +1
, 𝑢3 𝑥2
∫(𝑢2 + 1) 𝑑𝑢 = ∫(−𝑥 + 2) 𝑑𝑥; 3
+ 𝑢 = − +2x+C
2
1 𝑥2
𝑥 3 𝑦 3 + 𝑥𝑦 = − + 2𝑥 + 𝐶; 2𝑥 3 𝑦 3 + 6𝑥𝑦 + 3𝑥 2 − 12𝑥 = 𝐶
3 2
Ejemplo 15.
Halle la solución de la siguiente diferencial reducible a variables separables.
𝑑𝑦 2𝑥 − 3𝑦 + 1
− =0
𝑑𝑥 −6𝑥 + 9𝑦 + 2
Solución.
𝑑𝑦 2𝑥−3𝑦+1 𝑑𝑦 2𝑥−3𝑦+1 𝑑𝑦 2𝑥−3𝑦+1 𝑑𝑢 𝑑𝑦
− = 0; = ; = cambio de variable 𝑢 = 2𝑥 − 3𝑦; =2−3 ;
𝑑𝑥 −6𝑥+9𝑦+2 𝑑𝑥 −6𝑥+9𝑦+2 𝑑𝑥 −3(2𝑥−3𝑦)+2 𝑑𝑥 𝑑𝑥
𝑑𝑢
−2 𝑑𝑦
𝑑𝑥
=
−3 𝑑𝑥
𝑑𝑢
−2 𝑢+1 𝑑𝑢 −3𝑢−3 𝑑𝑢 −3𝑢−3 𝑑𝑢 −3𝑢−3−6𝑢+4
𝑑𝑥
= ; −2= ; = + 2; =
−3 −3𝑢+2 𝑑𝑥 −3𝑢+2 𝑑𝑥 −3𝑢+2 𝑑𝑥 −3𝑢+2
𝑑𝑢 −9𝑢+1 −3𝑢+2 −3𝑢+2
= ; 𝑑𝑢 = 𝑑𝑥; ∫ 𝑑𝑢 = ∫ 𝑑𝑥 dividiendo
𝑑𝑥 −3𝑢+2 −9𝑢+1 −9𝑢+1
5
1 1 5
∫ (3 + −9𝑢+1
3
) 𝑑𝑢 = ∫ 𝑑𝑥 ;
3
𝑢−
27
ln(−9𝑢 + 1) = 𝑥 + 𝐶 ;
1 5
(2𝑥 − 3𝑦) − ln(−9(2𝑥 − 3𝑦) + 1) = 𝑥 + 𝐶
3 27
1 5
𝑦+ 𝑦+ ln(1 + 27𝑦 − 18𝑥) = 𝐶
3 27
Ejemplo 16.
Halle la solución de la siguiente ecuación:
𝑦 ′ √𝑥 + 𝑦 + 1 = 𝑥 + 𝑦 − 1
Solución.
𝑦 ′ √𝑥 + 𝑦 + 1 = 𝑥 + 𝑦 + 1 − 1 − 1; si 𝑢 = 𝑥 + 𝑦 + 1 entonces;
𝑑𝑦 𝑥+𝑦−2 𝑑𝑢 𝑑𝑦 𝑑𝑦 𝑑𝑢
= si =1+ ; = − 1 sustituyendo y separando las variables
𝑑𝑥 √𝑥+𝑦+1 𝑑𝑥 𝑑𝑥 𝑑𝑥 𝑑𝑥
𝑑𝑢 𝑢−2 𝑑𝑢 𝑢+√𝑢−2
= +1; =
𝑑𝑥 √𝑢 𝑑𝑥 √𝑢
√𝑢 √𝑢
𝑑𝑢 = 𝑑𝑥 aplicando el operador de integración ∫ 𝑑𝑢 = ∫ 𝑑𝑥 si
𝑢+√𝑢−2 𝑢+√𝑢−2
𝑧 = √𝑢 ; 𝑢 = 𝑧 2 ; aplicando el operador de derivación en ambos lados de la igualdad
𝑑 𝑑𝑢 𝑑𝑧
(𝑢 = 𝑧 2 ); = 2𝑧 ; 𝑑𝑢 = 2𝑧𝑑𝑧 sustituyendo
𝑑𝑥 𝑑𝑥 𝑑𝑥
2𝑧 2 𝑑𝑧 𝑧2 −𝑧+2
∫ 𝑧 2+𝑧−2 = ∫ 𝑑𝑥 + 𝐶; dividiendo 𝑧 2 +𝑧−2
=1+
𝑧 2 +𝑧−2
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller jocelynmarcial30. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $9.49. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
76799 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now