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
Ronch [10]
3 years ago
11

Given a second order linear homogeneous differential equation a2(x)y′′+a1(x)y′+a0(x)y=0 we know that a fundamental set for this

ODE consists of a pair linearly independent solutions y1,y2. But there are times when only one function, call it y1, is available and we would like to find a second linearly independent solution. We can find y2 using the method of reduction of order. First, under the necessary assumption the a2(x)≠0 we rewrite the equation as y′′+p(x)y′+q(x)y=0 p(x)=a1(x)a2(x), q(x)=a0(x)a2(x), Then the method of reduction of order gives a second linearly independent solution as y2(x)=Cy1u=Cy1(x)∫e−∫p(x)dxy21(x)dx where C is an arbitrary constant. We can choose the arbitrary constant to be anything we like. One useful choice is to choose C so that all the constants in front reduce to 1. For example, if we obtain y2=C3e2x then we can choose C=1/3 so that y2=e2x. Given the problem y′′−4y′+29y=0 and a solution y1=e2xsin(5x) Applying the reduction of order method we obtain the following y21(x)=Cy1u= e^(4x)sin^2(5x) p(x)= -4 and e−∫p(x)dx= x So we have ∫e−∫p(x)dxy21(x)dx=∫ 1 dx= x Finally, after making a selection of a value for C as described above (you have to choose some nonzero numerical value) we arrive at y2(x)=Cy1u= So the general solution to y′′−5y′+4y=0 can be written as y=c1y1+c2y2=c1 +c2
Mathematics
2 answers:
maks197457 [2]3 years ago
8 0

Answer:

  • eˆ(4*x/5)
  • -20/25
  • eˆ(0.8*x)
  • eˆ(4*x/5)/[eˆ(4*x/5)]
  • x
  • x*eˆ(2*x/5)
  • eˆ(0.4*x)
  • x*eˆ(2*x/5)

ad-work [718]3 years ago
4 0

The question is not clear. What is clear is that you are talking about solving differential equations using the method of reduction of order.

I will explain this method by solving the equation

y''- 4y' + 29 = 0

with y1 = e^(4x).

This would further help you to solve your problem if it is not in this question.

Step-by-step explanation:

Given the differential equation:

y''- 4y' + 29 = 0

with y1 = e^(4x)

To find the other solution using the method of reduction of order, we assume the second solution to be of the form

y2 = uy1 = ue^(4x)

Since this solution, just like the given solution, satisfies the given differential equation, then

y2'' - 4y2' + 29 = 0

y2' = u'e^(4x) + 4ue^(4x)

y2'' = u''e^(4x) + 4u'e^(4x) + 4u'e^(4x) + 16ue^(4x)

= u''e^(4x) + 8u'e^(4x) + 16ue^(4x)

y2'' - 4y2' + 29 = [u''e^(4x) + 8u'e^(4x) + 16ue^(4x)] - 4[u'e^(4x) + 4ue^(4x)] + 29

= u''e^(4x) + 4u'e^(4x) + 29 = 0

u'' + 4u' = -29e^(-4x)

Let w = u', then w' = u''

So

w' + 4w = -29e^(-4x)

Multiply both sides by the integrating factor e^(4x)

w'e^(4x) + 4we^(4x) = -29

d(we^(4x)) = -29

Integrating both sides

we^(4x) = -29x

w = -29xe^(4x)

But w = u'

u' = -29xe^(4x)

Integrating this, we have

u = (29/16)(1 - 4x)e^(4x) + C

Since y2 = uy1

The second solution is now

y2 = (29/16)(1 - 4x)e^(8x) + Ce^(4x)

You might be interested in
Write an expression that is equivalent to 3/4 a + 2/3 - 1/2 a - 1/3 + 1/2 a​
Maksim231197 [3]
Hi there is this 0pp called Math-way that will help
7 0
2 years ago
Find the sum.<br> 7/30 + 9/20 PLS
e-lub [12.9K]

Answer:

41/60

Step-by-step explanation:

Convert the fractions into like denominators: 7/30 = 140/600, 9/20 = 270/600

Add: 140/600+270/600 = 410/600

Divide into a smaller fraction: 41/60

3 0
3 years ago
Is this a function or no?
IgorLugansk [536]

Answer:

No

Step-by-step explanation:

Positives cannot point at negatives I think.

5 0
2 years ago
Hiroto's texting plan costs $20 per month, plus $0.05 per text
Marta_Voda [28]
We can’t see the graph
8 0
2 years ago
Read 2 more answers
Expand and simplify x(x − 3)(x + 5)
konstantin123 [22]

Answer: x³+2x²-15x

Step-by-step explanation:

1. Expand

(x²-3x)(x+5)

2. Use the FOIL method: (a+b)(c+d)=ac+ad+bc+bd

x³+5x²−3x²−15x

3. Collect like terms

x³+(5x²−3x​²)−15x

4. Simplify

x³+2x²-15x

Hope this helps, have a BLESSED and wonderful day! ;-)

6 0
2 years ago
Other questions:
  • what is the mean of the numbers 5433221 round to the nearest hundredth if necessary. NEED ANSWER ASAP
    6·1 answer
  • The circle is inscribed in triangle AEC. A circle is inscribed in triangle A E C. Points B, F, and D of the circle are on the si
    7·2 answers
  • Liam works as a groundskeeper at a racetrack, where he is part of a bigger crew. When x people are working on the crew, it takes
    15·1 answer
  • The line plot shows students favorite pizza toppings. Which can you find using the line plot: the median, mode, range, or outlie
    11·1 answer
  • Suppose there are 7 roads connecting town a to town b and 8 roads connecting town b to town c. In how many ways can a person tra
    7·1 answer
  • 80 times 30 plus 60 equals
    7·2 answers
  • Helppppppppppppppppppppp
    12·1 answer
  • A circle has a circumference of 1,133.54 units.What is the diameter of the circle?
    10·1 answer
  • Write an equation that represents the line
    9·1 answer
  • Sarah has a picture that fits a 3 inch wide by 5 inch long picture frame. She wants to make the picture proportionately bigger s
    5·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!