1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Sergio [31]
3 years ago
8

Use an appropriate substitution to solve the equation

Mathematics
1 answer:
Yuri [45]3 years ago
4 0
y'-\dfrac8xy=\dfrac{y^5}{x^{20}}

This ODE is in standard Bernoulli form, so we can divide through both sides by y^5 to get

y^{-5}y'-\dfrac8xy^{-4}=\dfrac1{x^{20}}

Substitute z=y^{-4}, so that z'=-4y^{-5}y'. Then the ODE can be written as a linear one in z:

-\dfrac14z'-\dfrac8xz=\dfrac1{x^{20}}
z'+\dfrac{32}xz=-\dfrac4{x^{20}}

Multiplying both sides by x^{32} gives

x^{32}z'+32x^{31}z=-4x^{12}
(x^{32}z)'=-4x^{12}
x^{32}z=\displaystyle-4\int x^{12}\,\mathrm dx
x^{32}z=-\dfrac4{13}x^{13}+C
z=-\dfrac4{13x^{19}}+\dfrac C{x^{32}}

Back-substitute to solve for y:

\dfrac1{y^4}=-\dfrac4{13x^{19}}+\dfrac C{x^{32}}
y^4=\dfrac1{Cx^{-32}-\frac4{13}x^{-19}}
y=\left(\dfrac{x^{32}}{C-\frac4{13}x^{13}}\right)^{1/4}
y=\dfrac{x^8}{(C-\frac4{13}x^{13})^{1/4}}

Given that y(1)=1, you have

1=\dfrac{1^8}{(C-\frac4{13}1^{13})^{1/4}}
1=\dfrac1{(C-\frac4{13})^{1/4}}
1=\left(C-\dfrac4{13}\right)^{1/4}
1=C-\dfrac4{13}
C=\dfrac{17}{13}

so that the particular solution is

y=\dfrac{x^8}{(\frac{17}{13}-\frac4{13}x^{13})^{1/4}}
You might be interested in
Carl runs around a track in 1.17 minutes. How long would it take him to run 1.5 laps at this rate? [Round to two places
Andru [333]
Multiply:
1.17 x 1.5 = 1.755
 rounded to 2 places = 1.76 minutes
6 0
3 years ago
Read 2 more answers
Which number line best shows the position of square root of 7?
Serggg [28]

Step-by-step explanation:

First off, you want to find the whole number in square root form

1 = √1

2 = √4

3 = √9

4 = √16

5 = √25

You want to find which to numbers √7 is in between

7 is between 4 and 9 so the √7 is in between √4 and √9 which means that √7 is between 2 and 3

The second one is your answer

I hope this helps!!!

8 0
3 years ago
What is the measure of YB?
pishuonlain [190]
The formula is
YC x YD = AY x YB

fill in what we know:

18 x 6 = 9 x YB

108 = 9 x YB

YB = 108 / 9

YB = 12
5 0
3 years ago
I need this answer ?
FromTheMoon [43]
Are you sure you need help with that easy problem read it slowly then ask
7 0
3 years ago
The prime factorization of 16x5
RoseWind [281]

Answer:

well first 16x5=80

2 * 2 * 2 * 2 * 5

Step-by-step explanation:

Hope that helps! :)

6 0
3 years ago
Other questions:
  • You purchase a burrito for $8.39 and a juice for $2.25. You pay for both items with a $10 bill and a $5 bill. Complete a verbal
    10·1 answer
  • Determine if the two lines are perpendicular. Explain how you know by calculating the slopes and comparing the slopes in 2-3 sen
    14·1 answer
  • Suppose the radius of a circle is 4. What is its area?
    9·1 answer
  • (CEFET – MG) Sendo , a metade do valor de y valeA)
    14·1 answer
  • Superman was said to be "faster than a speeding bullet." A Winchester rifle bullet travels at a speed of 1,924 miles per hour (m
    6·2 answers
  • What equation represents a line which is perpendicular to x=0
    13·1 answer
  • What is the area of a trapezoid with height 5m and bases 8m and 1m.
    13·2 answers
  • An apple cost Php x per piece. How many will 3 dozen cost?
    7·1 answer
  • What do I do I don’t know the answer
    7·1 answer
  • Find all complex solutions of the equation 2^2 − 6 + 49 = 0. Express complex numbers in the form + .
    14·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!