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
daser333 [38]
2 years ago
7

Can someone help me solving this differential equation?

Mathematics
1 answer:
maria [59]2 years ago
8 0

Use reduction of order. Given a solution y_1(x) = x^2, look for a second solution of the form y_2(x) = y_1(x)v(x).

Compute the first two derivatives of y_2(x):

y_2 = x^2v \\\\ {y_2}' = x^2v' + 2xv \\\\ {y_2}'' = x^2v''+4xv' + 2v

Substitute them into the ODE:

x^4 (x^2v'' + 4xv' + 2v) + x^3 (x^2v' + 2xv) - 4x^2 (x^2v) = 1 \\\\ x^6v'' + 5x^5v' = 1

Now substitute w(x) = v'(x) and you end up with a linear ODE:

x^6w'+5x^5w=1

Multiply through both sides by \frac1x (if you're familiar with the integrating factor method, this is it):

x^5w'+5x^4w = \dfrac1x

Bear in mind that in order to do this, we require x\neq0. Just to avoid having to deal with absolute values later, let's further assume x>0.

Notice that the left side is the derivative of a product,

\left(x^5w\right)' = \dfrac1x

Integrate both sides with respect to x :

x^5w = \displaystyle \int\frac{\mathrm dx}x \\\\ x^5w = \ln(x) + C_1

Solve for w(x) :

w = \dfrac{\ln(x)+C_1}{x^5}

Solve for v(x) by integrating both sides:

v = \displaystyle \int \frac{\ln(x)+C_1}{x^5} \,\mathrm dx

Integrate by parts:

\displaystyle f = \ln(x) + C_1 \implies \mathrm df = \frac{\mathrm dx}x \\\\ \mathrm dg = \frac{\mathrm dx}{x^5} \implies g = -\frac1{4x^4} \\\\ \implies v = -\frac{\ln(x)+C_1}{4x^4} + \frac14 \int \frac{\mathrm dx}{x^5} \\\\ v = -\frac{\ln(x)+C_1}{4x^4} - \frac1{16x^4} + C_2 \\\\ v = -\frac{4\ln(x)+C_1}{16x^4}+C_2

Solve for y_2(x) :

\displaystyle \frac{y_2}{x^2} = -\frac{4\ln(x)+C_1}{16x^4}+C_2 \\\\ y_2 = -\frac{4\ln(x)+C_1}{16x^2} + C_2x^2

But since y_1(x)=x^2 is already accounted for, the second solution is just

\displaystyle y_2 = -\frac{4\ln(x)+C_1}{16x^2}

Still, the general solution would be

\displaystyle \boxed{y(x) = -\frac{4\ln(x)+C_1}{16x^2} + C_2x^2}

You might be interested in
1)4/9 ÷ 2/3
den301095 [7]
1. 2/3. Flip 2/3 into 3/2 and then multiply and simplify.
2. 30/91. Flip 7/6 to 6/7 and then multiply. You cannot simplify the fraction.
3. 26/27. Flip 9/10 to 10/9 and then multiply. You cannot simplify the fraction.
4. 44.2. Multiply it as if there was no decimal. Then count the number of digits after the decimal in each factor. Then put the same number of digits behind the decimal in the product.
5. 98.75. Multiply it as if there was no decimal. Then count the number of digits after the decimal in each factor. Then put the same number of digits behind the decimal in the product.
6. 3.36. Multiply it as if there was no decimal. Then count the number of digits after the decimal in each factor. Then put the same number of digits behind the decimal in the product.
7. 2. Multiply the divisor by as many 10’s as necessary until you get a whole number. Remember to multiply the dividend by the same number of 10’s. Then divide it normally.
8. 10.93 (rounded). Multiply the divisor by as many 10’s as necessary until you get a whole number. Remember to multiply the dividend by the same number of 10’s. Then divide it normally. I rounded it to the hundredth.
Hope this helps!
3 0
2 years ago
I WILL GIVE 20 POINTS TO THOSE WHO ANSWER THIS QUESTION RIGHT NOOOO SCAMS AND EXPLAIN WHY THAT IS THE ANSWER. Find the area of t
dem82 [27]

Answer:

easy 230-80 which is 150 so imagine a triangle with base 150 and hight 130-60=70 so times for area 150×70÷2 because it's triangle which is <em>5</em><em>2</em><em>5</em><em>0</em> so plus that with 130x80 for rectangle which is 10400 and add so it's 15650

8 0
2 years ago
Read 2 more answers
X^3 ÷ x^5<br> a)x^8<br> b)x^2<br> c)x^-8<br> d)x^-2
tatiyna

Answer:

D....................

Step-by-step explanation:

7 0
2 years ago
Balance the following redox reaction by ion electron method.<br><br>I2+ HNO3 = HI03+ NO2+ H20​
Ad libitum [116K]

Answer:

BRO I DONT KNOW SORY I NEED POINTS

Step-by-step explanation:

7 0
3 years ago
A chest shaped as a rectangular prism can hold 432 wooden cube blocks with edge lengths of 1/3 ft.
muminat

Answer:

16 ft³

Step-by-step explanation:

Volume of 1 cube = 1/3 x 1/3 x 1/3 = 1/27 ft³

Volume of 432 cubes = 432 x 1/27 = 16 ft³

7 0
2 years ago
Read 2 more answers
Other questions:
  • Halla dos numeros consecutivos cuyo producto sea 4224.
    10·1 answer
  • The point-slope form of the equation of the line that passes through (–5, –1) and (10, –7) is y plus 7 equals negative StartFrac
    9·2 answers
  • Has a center at (2,-1) and contains the point (4,5)
    12·1 answer
  • n the image, two circles are centered at A. The circle containing B was dilated to produce the circle containing B′. What is the
    5·1 answer
  • You are considering changing insurance companies. Your current company charges you $1500 per year for homeowners insurance plus
    8·1 answer
  • Solve following equations. <br> -2 (5+x) = -22
    8·1 answer
  • Help me please this is due tommorow
    6·1 answer
  • Two ingredients in a recipe are 2 1/2 cups of flour and 1 1/2 cups of sugar. of june keeps the proportion the same, how many cup
    10·1 answer
  • What is the slope of the lines that are parallel to the line defined by the equation y = -x – 4?
    13·1 answer
  • Why is the solution of 3x + 5 = 2x - 7
    6·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!