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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}

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F(x) = 2x + 3g(x) = 2x² + 6x + 13Find: (gºf)(x)
Butoxors [25]

Answer:

The function (gof)(x) is;

(g\circ f)(x)=8x^2+36x+49

Explanation:

Given the functions;

\begin{gathered} f(x)=2x+3 \\ g(x)=2x^2+6x+13 \end{gathered}

Solving for the function;

(g\circ f)(x)=g(f(x))

so, we have;

\begin{gathered} g(f(x))=2(f(x))^2+6(f(x))+13 \\ g(f(x))=2(2x+3)^2+6(2x+3)+13 \\ g(f(x))=2(4x^2+12x+9)^{}+6(2x)+6(3)+13 \\ g(f(x))=8x^2+24x+18^{}+12x+18+13 \\ g(f(x))=8x^2+24x^{}+12x+18+18+13 \\ g(f(x))=8x^2+36x+49 \end{gathered}

Therefore, the function (gof)(x) is;

(g\circ f)(x)=8x^2+36x+49

5 0
10 months ago
**The given angles are (-1+38x) and (36x+3)".<br><br> x=
topjm [15]

Answer:

x = 2

Step-by-step explanation:

these angles are alternate-interior angles which are congruent

-1 + 38x = 36x + 3

38x = 36x + 4

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x = 2

7 0
2 years ago
Nadia is mountain climbing. She started at an altitude of 19.26 feet below sea level and then changed her altitude by climbing a
lesya692 [45]

Answer:

5,418.2 feet

Step-by-step explanation:

Nadia is carrying out mountain climbing.

She started climbing the mountain at an altitude of 19.26 feet below the sea level.

Nadia changed her altitude by climbing a total of 5,437.8 feet from her starting position.

Therefore, Nadia's altitude at the end of her climb can be calculated as follows

= 5,437.8-19.6

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Hence Maria's altitude at the end of her climb is 5,418.2 feet

3 0
3 years ago
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ella [17]
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Adding 4x to other side
2y=10+4x
Then divide by 2
Y=10/2 +4/2x
Simplify
Y=5+2x
I rearranged so x is first so your answer would be: y=2x+5
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He is 15 years old this is 6 years younger than his sister Victoria write and solve a subtraction equation to find Victorious ag
Vedmedyk [2.9K]
15+6=21
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