CauchyEuler

CauchyEuler - Cauchy-Euler Equations The Cauchy-Euler...

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Cauchy-Euler Equations The Cauchy-Euler equation has the form a 0 x n y n ( ) + a 1 x n ! 1 y n ! 1 ( ) + ..... + a n ! 1 x " y + a n y = b(x) where a 0 , a 1 , …. , a n are constants. Each term contains x k y (k) . The transformation x = e t reduces the equation to a linear O.D.E. with constant coefficient in the variable t. Notice that we assume x > 0, and t = ln x. Using chain rule, since y is function of t through x: dy dt = dy dx dx dt = dy dx e t = dy dx x then x dy dx = dy dt Now, d 2 y dt 2 = d dt dy dt ! " # $ % = d dt x dy dx ! " # $ % = dx dt dy dx + x d dt dy dx ! " # $ % = e t dy dx + x d 2 y dx 2 dx dt = x dy dx + x 2 d 2 y dx 2 = dy dt + x 2 d 2 y dx 2 then x 2 d 2 y dx 2 = d 2 y dt 2 dy dt and d 3 y dt 3 = d dt d 2 y dt 3 ! " # $ % = d dt x dy dx + x 2 d 2 y dx 2 ! " # $ % = e t dy dx + x d 2 y dx 2 dx dt + 2e 2t d 2 y dx 2 + x 2 d 3 y dx 3 e t = x dy dx + x 2 d 2 y dx 2 + 2x 2 d 2 y dx 2 + x 3 d 3 y dx 3 = dy dt + 3x 2 d 2 y dx 2 + x 3 d 3 y dx 3 = dy dt + 3 d 2 y dt 2 3 dy dt + x 3 d 3 y dx 3 then x 3 d 3 y dx 3 = d 3 y dt 3 3 d 2 y dt 2 + 2 dy dt Following the model, you can compute the higher order derivatives. Remark : For the second order equation, the transformation produces: ax 2 d 2 y dx 2 + bx dy dx + cy = g(x), x > 0 taking x = e t a d 2 y dt 2 + (b ! a) dy dt + cy = g(e t )
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Example : 1) x 2 y’’ – 2xy’ + 2y = x
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CauchyEuler - Cauchy-Euler Equations The Cauchy-Euler...

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