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Unformatted text preview: Problem 1 (4+4+4+13) a. Discuss the roles of the following methods in solving linear ﬁrst order differential
equations including the kind of problems where these methods are applicable i) Method of variation of parameters ii) Method of reduction of order iii) Wronskian of two functions b. If y1 (t) = 1/ t, t > O is one of the ﬁmdamental solutions of the differential equation
tzy"+31y'+y=0, t>0
Find the other fundamental solution using the method of reduction of order. Also show that these solutions are fundamental solution of the differential equation. Solution Consider a linear, second order differential equation y” + p(x)y' + q(x) = g(x) . The general
solution can be written as y(x) = c1 y1(x) + czy2 (x) + yp (x) where yl (x) and y2 (x) are two linearly independent or fundamental solutions of the homogeneous equation
y” + p(x)y' + q(x) = 0 while yp (x) is any solution or particular solution of the non homogeneous equation y" + p(x) y’ + q(x) = g(x) . Various methods are available to ﬁnd the
solutions yl (x), y2 (x) and yp (x) i) Method of Variation of Parameters Application: This method is used to ﬁnd y p (x) When both fundamental solutions
y1(x) and y2 (x) are known. Method: We write the particular solution yp(x) as yp(x) =ul(x)y1(x)+u2(x)y2(x)
where u1 (x) and 2.42 (x) are two unknown functions of x . These functions are
determined from the condition that yp (x) is a solution of the nonhomogeneous
equation. Substituting in the differential equation, we can show that WP}! y,2(t)g(t)dt V, , 1420):"! x1(t)g(t)dt I y1(t)y2(t)"y2(t)y1(t) y1(t)y2(t)_y2(t)y1(t) Limitation: None as long as the integrals exist Method of Reduction of Order Application: This method is used to ﬁnd the other fundamental solution y2 (x) when
one of the fundamental solution y2 (x) is known. Method: We write y2 (x) as y2 (x) = v(x) y1 (x) .Substituting in the homogeneous
differential equation, we can show that x —]p(t)dt
DOC) = I e ds (y1(s))2 Limitation: None as long as the integrals exist
Wronskian of Two Functions Application: This method is used to test the linear independence of two ﬁmdamental
solutions y1 (x) and y2 (x). This test ensures that two solutions of the homogeneous equations can be used as ﬁmdamental solutions Method: We define the Wronskian W[ y1 (x), y2 (x)] = W(x) of two given ﬁmctions as W(x) = y1(x)y;(x) y2(x)y1'(x) . For y1(x) and y2(x) to constitute a set of
fundamental solutions, W(x) ¢ 0 within the domain of existence of the differential
equation. Limitation: None (b) In this case y1 (t) =% and p(t) = Hence
t ]p(x)dxd 1
t
y2(t) = y1(t) e—z—s = i, t > 0
{y1(s)} t In this case the Wronskian W(t) = y1 (t) 5’2 (t) — y1 (t) 5’2 (t) = Z13— ¢ 0, for t > 0. Hence yl (t) and y2 (t) are fundamental solutions. Problem 2 (5+10+5) Consider the displacement u(t) of dynamic system deﬁned by mass m , spring constant k
and damping coefﬁcient c , and governed by the differential equation mii+c11+ku=F(t)=E,sin(wt), m,c,k>0 a) Deﬁne the steady state response of the system. 2
b) Show that when G 2km
when the ﬁequency a) of the external force F (t) is equal to a) 1—62 wJZ
0V 2km, 0 m c) In an experimental design, it is desired to have a maximum steady state amplitude at a driving frequency of 31—5 times the natural frequency (00. If the spring is <1, the amplitude of the steady state response is maximum stretched 10 cm by a force of 3 newtons, and m = 2 kg , ﬁnd the damping
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a” in!) Problem 3 (5+5+8+4+3) 1. What are the characteristic differences in the solutions of a linear, second order
differential equation (LSDE) near an ordinary point and a regular singular point?
2. Consider the power series representation of two fundamental solutions y1 (x) and y2 (x) near a regular singular point x = x0 of a LSDE. Which one of the
following statements is valid and why? Explain your answer. 2a. lim yl (x) and lim y2 (x) are always ﬁnite x—)xo 2b. lim y1 (x) and lim y2 (x) may not be ﬁnite x—~)xo 3a. Derive the general solution of the Euler equation for x > 0
xzy”+2xy'—6y = 0 3b. Using the general solution, derive the complete solution that satisﬁes the
boundary conditions y(1) = l, y'(l) = —1
Is the solution bounded near x=0? 3c. The displacement y(x) of a physical problem is governed by the Euler equation in 3a. If the solution near x = 0 is bounded, write down the general solution of the
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of the following differential equation (1+x2)y'—4xy'+6y=0 aboutx= 0 b. Using the recurrence relation, ﬁnd the power series expansions of two
fundamental solutions of the differential equation about x=0. c. What are the radii of convergence of the ﬁmdamental solutions. 40‘ x20 ‘3“‘~ W‘:K&Y3 Vazvv‘ ﬂ 4kg C.(:G¢ﬂm’"/Q' Herc; M ?‘wtx AMen 3&QM'4Wr (an... (3' «’71?MA7A'J M
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 Spring '08
 KING
 yl, fundamental solutions

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