du dy dt = dx dt dx Bonus Question. Find the general solution of t2y" - 2ty' + 2y = 0 using the change of variables t = e. In other words, let v(x) = y(t) = y(e), and use the chain rule to calculate v'(x) and v"(x) in terms of t and y (using t e again). (Caution: finding v"(x) will involve using both the product and chain rules.) Use these expressions to convert the ODE for y into an ODE for v(x). This should be a second-order linear homogeneous equation with constant coefficients. Solve this equation for v(x) and translate your answer back to y(t) using x In t. - (This trick works for any 2nd-order equation of the form At²y" + Bty' + Cy = 0, which is called a Cauchy-Euler equation.)
du dy dt = dx dt dx Bonus Question. Find the general solution of t2y" - 2ty' + 2y = 0 using the change of variables t = e. In other words, let v(x) = y(t) = y(e), and use the chain rule to calculate v'(x) and v"(x) in terms of t and y (using t e again). (Caution: finding v"(x) will involve using both the product and chain rules.) Use these expressions to convert the ODE for y into an ODE for v(x). This should be a second-order linear homogeneous equation with constant coefficients. Solve this equation for v(x) and translate your answer back to y(t) using x In t. - (This trick works for any 2nd-order equation of the form At²y" + Bty' + Cy = 0, which is called a Cauchy-Euler equation.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:=
dx
dt dx
Bonus Question. Find the general solution of t2y" - 2ty' + 2y = 0 using the change
of variables t = e. In other words, let v(x) = y(t) = y(e), and use the chain rule
du dy dt
to calculate v'(x) and v"(x) in terms of t and y (using t e again).
(Caution: finding v"(x) will involve using both the product and chain rules.) Use these
expressions to convert the ODE for y into an ODE for v(x). This should be a second-order
linear homogeneous equation with constant coefficients. Solve this equation for v(x) and
translate your answer back to y(t) using x In t.
-
(This trick works for any 2nd-order equation of the form At²y" + Bty' + Cy = 0, which
is called a Cauchy-Euler equation.)
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