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
tester [92]
3 years ago
9

Solve this problem by using variation of parameters method. y''-y=coshx.

Mathematics
1 answer:
Gekata [30.6K]3 years ago
3 0
y''-y=0\implies r^2-1=0\implies r=\pm1
\implies y_c=C_1e^x+C_2e^{-x}={C^*}_1\underbrace{\cosh x}_{y_1}+{C^*}_2\underbrace{\sinh x}_{y_2}

For the nonhomogeneous ODE

y''-y=\underbrace{\cosh x}_{f(x)}

we're looking for a particular solution of the form

y_p=u_1y_1+u_2y_2

where

u_1=-\displaystyle\int\frac{y_2(x)f(x)}{W(y_1(x),y_2(x))}\,\mathrm dx
u_2=\displaystyle\int\frac{y_1(x)f(x)}{W(y_1(x),y_2(x))}\,\mathrm dx

and W(y_1,y_2) is the Wronskian of the two fundamental solutions.

We have

W(y_1,y_2)=\begin{vmatrix}\cosh x&\sinh x\\\sinh x&\cosh x\end{vmatrix}=\cosh^2x-\sinh^2x=1

so we're left with

u_1=-\displaystyle\int\sinh x\cosh x\,\mathrm dx=-\dfrac12\cosh^2x
u_2=\displaystyle\int\cosh^2x\,\mathrm dx=\dfrac12x+\dfrac14\sinh2x

so that the particular solution is

y_p=-\dfrac12\cosh^3x+\dfrac12x\sinh x+\dfrac14\sinh x\sinh2x
y_p=-\dfrac12\cosh x+\dfrac12x\sinh x

As y_1 already accounts for the \cosh x term in y_p, we're left with the general solution

y=C_1\cosh x+C_2\sinh x+\dfrac12x\sinh x
You might be interested in
What two things make a graph proportional?​
sp2606 [1]
The y intercept and the gradient
6 0
3 years ago
Read 2 more answers
A ladder is leaning against a vertical wall which is 5m high. The top of the ladder slides all the way down the wall so that the
Lynna [10]

Answer:

Do using Pythagoras theorem... Also before doing a question of height and distance always try to make the figure

Step-by-step explanation:

Use the two triangles ABC and ABD to first get the base and find the height of the ladder

7 0
3 years ago
What is the difference of the polynomials?<br> (x2 - 6x + 2) - (3x? – 7x-8)
bagirrra123 [75]
The difference is 10.
4 0
2 years ago
I need help with this
Darya [45]

Answer:

76

Step-by-step explanation:

We know that the two angles must add up to 90 which means that

12+x-2=90

solve for x

14+x=90

x= 76

8 0
2 years ago
What is the answer this equation<br> 3/8 × 32/5
cupoosta [38]
96/40
simplified 12/5
7 0
3 years ago
Read 2 more answers
Other questions:
  • PQ and RS are two lines that intersect at point T, as shown below:
    11·2 answers
  • I need help with 3,4 and 5. Please.
    6·1 answer
  • Θ is in Quadrant III and cos^2 0= 1/4
    9·1 answer
  • What is 2,767,545 to the nearest ten
    6·1 answer
  • Tom wishes to purchase a property that has been valued at $300,000. He has $30,000 available as a deposit, and will require a mo
    13·1 answer
  • On average, a furniture store sells four card tables in a week. Assuming a Poisson distribution for the weekly sales, the probab
    15·1 answer
  • PLEASE HELP: A medication 60 mg is ordered. the vial is labeled 1 gm/ 10 ml. How many ml would you give?
    6·1 answer
  • Which shows the best path to conert 100 liters to pints
    6·2 answers
  • The graph shows the motion of two cars starting at different places on a highway. Their speeds can be compared by comparing the
    9·1 answer
  • A cash card has a starting value of $25. If the card is not used within the first year of its purchase, the value on the card be
    7·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!