Problem 1.2. Consider again the exact model from the previous problem and write 1 YMɛ² (1 + εr)2¹ R It can be shown that the solution r(t) = r(t, e) to the above initial conditions is C (with respect to both t and e). Show that 7 = h, 1² 27 2 +0(₁), RA YM R² r(t) = h-g(1-2 9= (Hint: Insert r(t, e) = ro(t) + r₁(t)e + r₂(t)e² +r3(t)e³ + 0(4) into the differential equation and collect powers of e. Then solve the corresponding differential equations for ro(t), ri(t), ... and note that the initial conditions follow from r(0, e) = h respectively r(0, c) = 0. A rigorous justification for this procedure will be given in Section 2.5.)
Problem 1.2. Consider again the exact model from the previous problem and write 1 YMɛ² (1 + εr)2¹ R It can be shown that the solution r(t) = r(t, e) to the above initial conditions is C (with respect to both t and e). Show that 7 = h, 1² 27 2 +0(₁), RA YM R² r(t) = h-g(1-2 9= (Hint: Insert r(t, e) = ro(t) + r₁(t)e + r₂(t)e² +r3(t)e³ + 0(4) into the differential equation and collect powers of e. Then solve the corresponding differential equations for ro(t), ri(t), ... and note that the initial conditions follow from r(0, e) = h respectively r(0, c) = 0. A rigorous justification for this procedure will be given in Section 2.5.)
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:Problem 1.2. Consider again the exact model from the previous problem
and write
YMɛ²
(1 + εr)2¹
It can be shown that the solution r(t) = r(t, e) to the above initial conditions
is C (with respect to both t and e). Show that
€ =
1
R
h, 1²
r(t)=h-g(1-2- +
R 2
(Hint: Insert r(t, e) = ro(t) + ri(t)e + r2(t)e² + r3(t)e³ + O(¹) into the
differential equation and collect powers of e. Then solve the corresponding
differential equations for ro(t), ri(t), ... and note that the initial conditions
follow from r(0, e) = h respectively r(0, e) = 0. A rigorous justification for
this procedure will be given in Section 2.5.)
-0(₁),
YM
R²
9 =
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