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DEFERENTIAL EQUATION exam, Exams of Computer Engineering and Programming

Stanford University Computer Engineering and Programming

A set of problems related to differential equations and integration by parts. It includes examples of eliminating arbitrary constants, separation of variables, and integration by parts. The document also shows how to solve specific differential equations using different methods. The problems are presented in a table format with their corresponding solutions. The document can be useful for students studying differential equations and integration by parts in mathematics or engineering courses.

Typology: Exams

2021/2022

Available from 02/19/2022

Pau5 🇵🇭

2 documents

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Differential

Equation

Discover Exams of Computer Engineering and Programming Stanford University

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Differential Equation

(^) Table of Contents
(^) Eliminating Arbitrary Constant
(^) No. 12 ..................................................................
No.25 ..................................................................
(^) Integration by Parts
(^) No. 3 ..................................................................
(^) No. 3 ..................................................................
(^) Separation of Variables
(^) No. 4 ..................................................................
(^) No. 12 .................................................................
(^) No. 18 .................................................................

25.) y = C1x^2 + C2e^2x (eq.1)

y’ = 2C1x + 2C2e^2x (eq.2)

y = C1x^2 + C2e^2x (eq.3)

-2 multiply to equation 1 then subtract to equation 2

-2y = - 2C1x^2 – 2C2e^2x

y’ = 2C1x^2 – 2C2e^2x
- y’ = 2C1x^2 – 2C2e^2x

y’-2y = C1 (2x – 2x^2)

2x-2x^2 2x – 2x^

(eq.4)

Y’-2y = C1 (2x – 2x^e) C1 = y’-2y 2x-2x^

Y’’- 2y’=C1(2-4x) (y” – 2y’) (2x – 2x^2) = (y’- 2y) (2 - 4x)

(t^4 𝑒^2 )/2 - (t 𝑡 𝑒^2 )/8 + ( 𝑡 𝑒^2 )/8 + C 𝑡

3.) ꭍydy/(y-1)^ U= y dy=dy Du= (y-1)^5dy y= - ¼(y-1)^ Let u = y – 1 u^5=7 u- u = 5 + 1 = - ¼(y-1)^

5 + 1 = 7 y= 1/(-4(x-1))^4 - ꭍ1 - 1/(-4(x-1)) dy y - 1/(-4(x-1))^4 - ꭍ1 - 1/(-4(x-1)) dy
Y /(4(x-1))^4 - ꭍ1 - 1/ -4(y-1)^4 dy Let u = y – 1 = 1 u^4 = u^- = u – 4+1/-4+1 = u-3 1/3(7-1)^ -3= = ¼ (1/3(y-1)^3) =1/4 – 1/3 (y-1)^3 = 1/3(y-1)^ y/4(y-1)^4 – 1/12(y-1)^3 + C

Y=(x/2)^2/ Sin y = cos x

C^2y^2 = 4 + e^2x