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
Colt1911 [192]
3 years ago
10

Consider the differential equation

B2%7D%20%2B6xy" id="TexFormula1" title=" x^{2} \frac{dy}{dx} =6 y^{2} +6xy" alt=" x^{2} \frac{dy}{dx} =6 y^{2} +6xy" align="absmiddle" class="latex-formula"> which may be considered either as a homogenous equation or as a Bernoulli equation.
If we make the substitution y(x)=xv(x) relevant to homogenous equations, we obtain \frac{dv}{dx}=
If we make the substitution z(x)= (y(x))^{-1} relevant to homogenous equations, we obtain \frac{dz}{dx}=
Using either (or both) of these methods, solve the initial value problem for the above equation where y(3)=6. Find the interval of validity of this solution.
Mathematics
1 answer:
Nutka1998 [239]3 years ago
8 0
As a Bernoulli equation:

x^2\dfrac{\mathrm dy}{\mathrm dx}=6y^2+6xy\iff x^2y^{-2}\dfrac{\mathrm dy}{\mathrm dx}-6xy^{-1}=6

Let z=y^{-1}\implies\dfrac{\mathrm dz}{\mathrm dx}=-y^{-2}\dfrac{\mathrm dy}{\mathrm dx}. The ODE becomes

-x^2\dfrac{\mathrm dz}{\mathrm dx}-6xz=6
x^6\dfrac{\mathrm dz}{\mathrm dx}+6x^5z=-6x^4
\dfrac{\mathrm d}{\mathrm dx}[x^6z]=-6x^4
x^6z=-6\displaystyle\int x^4\,\mathrm dx
x^6z=-\dfrac65x^5+C
z=-\dfrac6{5x}+\dfrac C{x^6}
y^{-1}=-\dfrac6{5x}+\dfrac C{x^6}
y=\dfrac1{\frac C{x^6}-\frac6{5x}}
y=\dfrac{5x^6}{C-6x^5}

With y(3)=6, we get

6=\dfrac{5(3)^6}{C-6(3)^5}\implies C=\dfrac{4131}2

so the solution is

y=\dfrac{5x^6}{\frac{4131}2-6x^5}=\dfrac{10x^6}{4131-12x^5}

which is valid as long as the denominator is not zero, which is the case for all x\neq\sqrt[5]{\dfrac{4131}{12}}.
You might be interested in
a parallelogram has an area of 108 square inches . the base of the parallelogram is 18 inches . explain how you could find the h
Paraphin [41]
The formula for perimeter is

Side a x2 + side b x2

So just multiply 18 by two, and subtract it from 108
4 0
3 years ago
If you pay $ 34.02 for a jacket that includes a 5% sales tax, what is the price of the jacket before the sales tax?
Jobisdone [24]
105% =$34.02
100% (cost without sales tax) = 34.02/1.05 =$32.40
3 0
3 years ago
Read 2 more answers
In right ∆ABC the altitude CH to the hypotenuse AB intersects angle bisector AL in point D. Find BC if AD = 8 cm and DH = 4 cm.
alina1380 [7]
Check the picture below.

notice that the triangle ADH, since the segment AL is an angle bisector, meaning  it cuts the angle A in two equal halves, then the triangle ADH is only using half of A.

5 0
3 years ago
Read 2 more answers
Calculate the number of roots of <br> 2sin^3x-5sin^2x+2sinx=0<br> in the interval 0&lt;=x&lt;=pi
natita [175]

Answer:

Which points are collinear?

G, A, and T

G, E, and T

A, T, and E

G, A, and E

Step-by-step explanation:

6 0
3 years ago
3. Multiply<br> (5 points)<br> O<br> 33<br> -6<br> 40<br> O<br> mo<br> O
ladessa [460]

Answer: -7920

Step-by-step explanation: 33x-6x40= -7920

5 0
3 years ago
Read 2 more answers
Other questions:
  • A box 5 cm long, 4 cm wide and 6 cm high would have what volume?
    5·2 answers
  • Which is the simplified version of 7x+12+3x=-6(x-7)+9x ?
    13·1 answer
  • Find the distance between the points (2,-1) and (-1,-4). Round your answer to the nearest hundredth.
    9·1 answer
  • When 47 is added to a number the result is 37 less than 7 times the number.
    13·1 answer
  • Need help figuring this equation 1/12c =.6
    6·1 answer
  • that one kilogram of rice weigh the same as 10 bags containing 100 grams of beans each do you agree with her why or why not
    9·1 answer
  • Which choice could be the equation of a line perpendicular to the line represented by this equation? y = 5x− 2
    6·1 answer
  • PLZ HELP List all the possible numbers of pennies that Gigi could have in order to have her pennies into rectangular array with
    15·1 answer
  • Angela plays soccer and golf for a total of 125 minutes every day. She plays soccer 45 minutes more than she plays golf.
    10·1 answer
  • Which expression is equivalent to 3^0x3^-3?
    11·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!