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anonymous
 one year ago
Question regarding one step in this problem
anonymous
 one year ago
Question regarding one step in this problem

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anonymous
 one year ago
Best ResponseYou've already chosen the best response.0\[x^2 y''  7xy' + 16; y_1 = x^4\]I have to set it in standard form following this rule\[y''+P(x)y'+Q(x)=0\]which results in\[y''\frac{ 7 }{ x } y' + \frac{ 16 }{ x^2 } y = 0\] How did the x and x^2 moved on the denominator?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0why is it that we have to divide it by x^2 after changing it to standard form?

dan815
 one year ago
Best ResponseYou've already chosen the best response.1you dicide to change it to standard form

thomas5267
 one year ago
Best ResponseYou've already chosen the best response.0That looks like it, unless the equation is inhomogeneous and you have to figure out the inhomogeneous part by the given solution.

dan815
 one year ago
Best ResponseYou've already chosen the best response.1divide*, you cannot have any f(x) multiypling your highest degree

thomas5267
 one year ago
Best ResponseYou've already chosen the best response.0\(x^2 y''  7xy' + 16=0\) is not in standard form because there is \(x^2\) in front of \(y''\) where in standard form there is nothing.

dan815
 one year ago
Best ResponseYou've already chosen the best response.1first solve the homogenous equation then use variation of parameters to get general solution to your non homogenous equation

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0so to set it to standard form whatever is in front of y'' has to be divided to just have y'' alone

thomas5267
 one year ago
Best ResponseYou've already chosen the best response.0\[ x^2 y''  7xy' + 16\\ y_1 = x^4 \] But plugging \(y_1\) into the above expression I get \(16x^4+16\).

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0oh no you have to do reduction of sides

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0\[y_2 = y_1 \int\limits_{}^{}\frac{ e ^{\int\limits_{}^{}P(x)dx} }{y_1^2 }\]

thomas5267
 one year ago
Best ResponseYou've already chosen the best response.0I am assuming \(y_1=x^4\) is a solution to the incomplete differential equation \(x^2 y''  7xy' + 16\). What does this equation equal to?

thomas5267
 one year ago
Best ResponseYou've already chosen the best response.0Or is \(y_1\) not a solution?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0The final answer is \[Y=C_1x^4 + C_2 x^4\ln(x)\]or simplified \[Y = x^4(C_1 + C_2 \ln(x))\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0using reduction of sides you find y_2

thomas5267
 one year ago
Best ResponseYou've already chosen the best response.0In other words, \(y_1=x^4\) should satisfy the equation \(x^2 y''  7xy' + 16=0\)? \[ x^2(x^4)''7x(x^4)'+16\\ 12x^428x^4+16\\ 16x^4+16\neq0 \]

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.1try to search for solution of this type: \[y = k{x^s},\quad s \in \mathbb{Z}\]

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.1\[\Large y = k{x^s},\quad s \in \mathbb{Z}\]

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.1after the substitution, you should get a quadratic equation for \(s\)

thomas5267
 one year ago
Best ResponseYou've already chosen the best response.0\[ x^2 y''  7xy' + 16=0 \text{ not }x^2 y''  7xy' + 16y=0 \] I got\[ r(r1)x^r7rx^r+16=0 \] after assuming \(y=x^r\).

thomas5267
 one year ago
Best ResponseYou've already chosen the best response.016 does not have y behind it in the differential equation. Not sure whether it is a typo or not.

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.1from first reply I see 16y/x^2

Michele_Laino
 one year ago
Best ResponseYou've already chosen the best response.1if we ahve only \(16\) and not \(16y\) then we have to apply my substitution above to the homogeneous equation only

thomas5267
 one year ago
Best ResponseYou've already chosen the best response.0Exactly the same equation on this website. http://tutorial.math.lamar.edu/Classes/DE/EulerEquations.aspx

thomas5267
 one year ago
Best ResponseYou've already chosen the best response.0If the equation is \[ x^2y''7xy'+16y=0 \] Divide the whole thing by \(x^2\) to get the standard form. Or just read the website.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0oh yeahI ate the y after the 16, sorry. I do undeerstand this though, just had doubt in the beginning

thomas5267
 one year ago
Best ResponseYou've already chosen the best response.0Y U ATE THE Y? YYYYY? Sorry.....
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