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
hjlf
3 years ago
8

2 points) Sometimes a change of variable can be used to convert a differential equation y′=f(t,y) into a separable equation. One

common change of variable technique is as follows. Consider a differential equation of the form y′=f(αt+βy+γ), where α,β, and γ are constants. Use the change of variable z=αt+βy+γ to rewrite the differential equation as a separable equation of the form z′=g(z). Solve the initial value problem y′=(t+y)2−1, y(3)=4.
Mathematics
1 answer:
Stells [14]3 years ago
3 0

y'=(t+y)^2-1

Substitute u=t+y, so that u'=y', and

u'=u^2-1

which is separable as

\dfrac{u'}{u^2-1}=1

Integrate both sides with respect to t. For the integral on the left, first split into partial fractions:

\dfrac{u'}2\left(\frac1{u-1}-\frac1{u+1}\right)=1

\displaystyle\int\frac{u'}2\left(\frac1{u-1}-\frac1{u+1}\right)\,\mathrm dt=\int\mathrm dt

\dfrac12(\ln|u-1|-\ln|u+1|)=t+C

Solve for u:

\dfrac12\ln\left|\dfrac{u-1}{u+1}\right|=t+C

\ln\left|1-\dfrac2{u+1}\right|=2t+C

1-\dfrac2{u+1}=e^{2t+C}=Ce^{2t}

\dfrac2{u+1}=1-Ce^{2t}

\dfrac{u+1}2=\dfrac1{1-Ce^{2t}}

u=\dfrac2{1-Ce^{2t}}-1

Replace u and solve for y:

t+y=\dfrac2{1-Ce^{2t}}-1

y=\dfrac2{1-Ce^{2t}}-1-t

Now use the given initial condition to solve for C:

y(3)=4\implies4=\dfrac2{1-Ce^6}-1-3\implies C=\dfrac3{4e^6}

so that the particular solution is

y=\dfrac2{1-\frac34e^{2t-6}}-1-t=\boxed{\dfrac8{4-3e^{2t-6}}-1-t}

You might be interested in
The scale of a map says that 3cm represents 8km. What distance on the map (in centimeters) represents an actual distance of 2 ki
makvit [3.9K]
X/2=3/8x
x=(2)3/8
x=6/8
x=3/4
x=0.75cm
5 0
3 years ago
Can someone please help me?
melamori03 [73]

Answer:

Answer is B

Step-by-step explanation:

3 0
2 years ago
Read 2 more answers
Felix says if 2/3 pan of lasagna had been left over from dinner and the padis family ate 1/2 of that for lunch, the answer would
olga_2 [115]

Answer:

He is right he can eat 2/6 f the whole pan of lasagna

Step-by-step explanation:

We had  2/3 pan of lasagna and pads family ate 1/2 of it, then

padis family ate  (2/3) /2     or  2/6    

2/6 is half of the total pan of lasagna, padis family ate half of 2/3 wich is 2/6 and left 2/6 for Felix

6 0
3 years ago
Plz help
Lina20 [59]
This question makes no sense
7 0
3 years ago
What do you multiple 24 with to get 120
ohaa [14]

Answer:

5

Step-by-step explanation:

I divided \frac{120}{24} = 5

3 0
3 years ago
Read 2 more answers
Other questions:
  • A 1,349 tv is on sale for 20% off. If sales tax is 8%, find the price of the tv including tax
    11·1 answer
  • 9x-2y=6<br> What is this equation in slope intercept form?
    7·2 answers
  • Which number is a common multiple of 7and 4<br><br><br> What is the least common multiple of 18and28
    12·1 answer
  • Find the value of 15/124•230/30 divided 230/124
    12·2 answers
  • Select all that apply.
    5·1 answer
  • △KLM has k&gt;l&gt;m. Which angle of the triangle may measure 60°?
    7·2 answers
  • Can anyone help with this?
    8·2 answers
  • Please help will give brainliest
    10·2 answers
  • Please help I need this done by tonight!!
    11·2 answers
  • Can someone please help me out?
    14·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!