Differential Equations, solve the following: y^(4) - y'' - 2y' +2y = 0

  • #1
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Homework Statement



Solve the following differential equations/initial value problem:

y^(4) - y'' - 2y' +2y = 0 Hint: e^-x sinx is a solution

Homework Equations



I was attempting to solve this problem by using a characteristic equation.

The Attempt at a Solution



y'''' -y'' -2y' + 2y = 0 --> r^4 -r^2 -2r +2 = 0

r^2(r^2 - 1) -2(r - 1) = 0

r^2 (r+1)(r-1) -2(r-1) = 0

(r^2(r+1) -2)(r-1) = 0

from this I get r = 1, but I know there are more solutions; How do I solve for the other solutions? And then write it as the solution?
 

Answers and Replies

  • #2
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Homework Statement



Solve the following differential equations/initial value problem:

y^(4) - y'' - 2y' +2y = 0 Hint: e^-x sinx is a solution

Homework Equations



I was attempting to solve this problem by using a characteristic equation.

The Attempt at a Solution



y'''' -y'' -2y' + 2y = 0 --> r^4 -r^2 -2r +2 = 0

r^2(r^2 - 1) -2(r - 1) = 0
This isn't a useful approach, IMO, unless you are able to factor the cubic polynomial.
komarxian said:
r^2 (r+1)(r-1) -2(r-1) = 0

(r^2(r+1) -2)(r-1) = 0

from this I get r = 1, but I know there are more solutions; How do I solve for the other solutions? And then write it as the solution?
Use the given hint. If ##y = e^{-x}\sin(x)## is a solution, then another solution is ##y = e^{-x}\cos(x)## is also a solution. This means that ##r_1 = -1 + i## and ##r_2 = -1 - i## are roots of some quadratic characteristic equation.
 
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  • #3
Ray Vickson
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Homework Statement



Solve the following differential equations/initial value problem:

y^(4) - y'' - 2y' +2y = 0 Hint: e^-x sinx is a solution

Homework Equations



I was attempting to solve this problem by using a characteristic equation.

The Attempt at a Solution



y'''' -y'' -2y' + 2y = 0 --> r^4 -r^2 -2r +2 = 0

r^2(r^2 - 1) -2(r - 1) = 0

r^2 (r+1)(r-1) -2(r-1) = 0

(r^2(r+1) -2)(r-1) = 0

from this I get r = 1, but I know there are more solutions; How do I solve for the other solutions? And then write it as the solution?
You have ##p(r) \equiv r^2 - r^2 -2r+2 = (r-1)(r^3+r^2-2).## Notice that the polynomial ##q(r) = r^3+r^2 -2## has a root ##r = 1##, so ##(r-1)## is a factor. That gives ##q(r) = (r-1)(r^2+2r+2) ##, hence ##p(r) = (r-1)^2 (r^2+2r+2).## Thus, ##r=1## is a double root of ##p(r)## and the other roots are found by solving a simple quadratic equation.
 
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  • #4
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This isn't a useful approach, IMO, unless you are able to factor the cubic polynomial.

Use the given hint. If ##y = e^{-x}\sin(x)## is a solution, then another solution is ##y = e^{-x}\cos(x)## is also a solution. This means that ##r_1 = -1 + i## and ##r_2 = -1 - i## are roots of some quadratic characteristic equation.
Thank you!
 
  • #5
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You have ##p(r) \equiv r^2 - r^2 -2r+2 = (r-1)(r^3+r^2-2).## Notice that the polynomial ##q(r) = r^3+r^2 -2## has a root ##r = 1##, so ##(r-1)## is a factor. That gives ##q(r) = (r-1)(r^2+2r+2) ##, hence ##p(r) = (r-1)^2 (r^2+2r+2).## Thus, ##r=1## is a double root of ##p(r)## and the other roots are found by solving a simple quadratic equation.
Thank you!! I get it now; gotta work on the algebra XD
I've got 99 problems and it's all algebra pretty much.
 
  • #6
Ray Vickson
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This isn't a useful approach, IMO, unless you are able to factor the cubic polynomial.

Use the given hint. If ##y = e^{-x}\sin(x)## is a solution, then another solution is ##y = e^{-x}\cos(x)## is also a solution. This means that ##r_1 = -1 + i## and ##r_2 = -1 - i## are roots of some quadratic characteristic equation.
You say that solving the characteristic equation may not be useful. However, sometimes that's all you can do. For example, the differential equation ##y''' - 2 y''
+3 y' - 4 = 0## has characteristic equation ##r^3-2 r^2+3 r - 4 = 0##. This can be solve usingCardano's formulas (for example), but that is really not a useful way to go: numerical solutions are far preferable. That gives the solutions as ##y_1 = e^{rx},## ## y_2 = e^{\alpha x} \cos(\beta x)## and ##y_3 = e^{\alpha x} \sin(\beta x),## where ##r \doteq 1.650629192,## ##\alpha \doteq 0.1746854042## and ##\beta \doteq 1.546868888.##
 
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  • #7
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You say that solving the characteristic equation may not be useful. However, sometimes that's all you can do.
I didn't say that solving the char. equation wasn't useful in this case, if the OP wasn't able to get other solutions out of the remaining cubic polynomial.
 

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