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
Olenka [21]
3 years ago
15

Solve the nonhomogeneous differential equation y′′+25y=cos(5x)+sin(5x). Find the most general solution to the associated homogen

eous differential equation. Use c1 and c2 in your answer to denote arbitrary constants. Enter c1 as c1 and c2 as c2.
Mathematics
1 answer:
Marrrta [24]3 years ago
6 0

Answer:

y(x)=c_1cos(5x)+c_2sin(5x)+0.1xsin(5x)-0.1xcos(5x)

Step-by-step explanation:

The general solution will be the sum of the complementary solution and the particular solution:

y(x)=y_c(x)+y_p(x)

In order to find the complementary solution you need to solve:

y''+25y=0

Using the characteristic equation, we may have three cases:

Real roots:

y(x)=c_1e^{r_1x} +c_2e^{r_2x}

Repeated roots:

y(x)=c_1e^{rx} +c_2xe^{rx}

Complex roots:

y(x)=c_1e^{\lambda x}cos(\mu x) +c_2e^{\lambda x}sin(\mu x)\\\\Where:\\\\r_1_,_2=\lambda \pm \mu i

Hence:

r^{2} +25=0

Solving for r :

r=\pm5i

Since we got complex roots, the complementary solution will be given by:

y_c(x)=c_1cos(5x)+c_2sin(5x)

Now using undetermined coefficients, the particular solution is of the form:

y_p=x(a_1cos(5x)+a_2sin(5x) )

Note: y_p was multiplied by x to account for cos(5x) and sin(5x) in the complementary solution.

Find the second derivative of y_p in order to find the constants a_1 and a_2 :  

y_p''(x)=10a_2cos(5x)-25a_1xcos(5x)-10a_1sin(5x)-25a_2xsin(5x)

Substitute the particular solution into the differential equation:

10a_2cos(5x)-25a_1xcos(5x)-10a_1sin(5x)-25a_2xsin(5x)+25(a_1xcos(5x)+a_2xsin(5x))=cos(5x)+sin(5x)

Simplifying:

10a_2cos(5x)-10a_1sin(5x)=cos(5x)+sin(5x)

Equate the coefficients of cos(5x) and sin(5x) on both sides of the equation:

10a_2=1\\\\-10a_1=1

So:

a_2=\frac{1}{10} =0.1\\\\a_1=-\frac{1}{10} =-0.1

Substitute the value of the constants into the particular equation:

y_p(x)=-0.1xa_1cos(5x)+0.1xsin(5x)

Therefore, the general solution is:

y(x)=y_c(x)+y_p(x)

y(x)=c_1cos(5x)+c_2sin(5x)+0.1xsin(5x)-0.1xcos(5x)

You might be interested in
Write a linear eqaution for the point (3,-5) and slope=-4
Citrus2011 [14]

Answer:

y = -4x + 7

Step-by-step explanation:

3 0
2 years ago
Read 2 more answers
There are 4 lunch choices in a school cafeteria: Salad, Pizza, Spaghetti, and Sub Sandwiches. The percentage of students that ch
Elina [12.6K]
I think it would be 300, because 15% of x=45, so .15x=45, x= 45/.15=300
8 0
3 years ago
Need help ASAP!!!!!!
Mnenie [13.5K]

Answer:

Mean = 2.95

Median = 3

Step-by-step explanation:

The mean of the number of visits of 40 adults to the gym is

\dfrac{0\cdot 4+1\cdot 3+2\cdot 12+3\cdot 5+4\cdot 8+5\cdot 5+6\cdot 2+7\cdot 1}{40}=\\ \\=\dfrac{3+24+15+32+25+12+7}{40}=\dfrac{118}{40}=2.95

The median of the data is the average number between 20th and 21th terms (because 40 is even number, so the middle term is the average between two middle terms).

20th term = 3 visits

21th term = 3 visits

Hence, median is 3 visits

6 0
3 years ago
3(b - 15) =9 solve for b
shepuryov [24]
B = 18 because 18-5 is equal to 3 and 3x3 is equal to 9.
4 0
3 years ago
433x3 how u start this off
den301095 [7]
Answer :
433 x 3
1299
4 0
3 years ago
Read 2 more answers
Other questions:
  • IF I travel (4+x) miles per hour for 2 hour and I travel the same distance at (4-x) miles per hour in 3 hours, what is x?
    14·1 answer
  • Whats -1 1/10 divided by -11/10
    11·2 answers
  • a house in Florida is priced at $155,000 the rent on a comparable houses $950 per month find the price to rent ratio​
    12·1 answer
  • Can someone explain this lol thanks Find the volume of the rectangular
    15·1 answer
  • On your first day in business, 126 people come to eat. They spend about $52 each. About how much money did you earn on your firs
    12·1 answer
  • Suppose you work for a marketing consulting firm and you are tasked to determine the proportion of Americans who respond positiv
    5·1 answer
  • Help would be much appreciated ​
    9·1 answer
  • Help ill give you 25 points
    9·1 answer
  • Can someone please solve this and save my life
    11·1 answer
  • Drop down answers are
    6·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!