M 427K-Vishik-Spring 2003-Test 2

M 427K-Vishik-Spring 2003-Test 2 - ~'\l.ll -:: .r;;;\'_...

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M 427K Midterm exam March 27 , 2003 NAM~~~~~~~~~~~~ Unique # 56045 STUDENT # _ Please show all work clearly and circle all answers. There are 5 problems: 20 points each. Find a fundamental set of solution of the differential equation (6) (4) II W · y +3y +3y ,0 . Using variation of parameters find the g~neral solution of rJ the ODE yl' - 16 y = g(x), + y = 0 where g(x) is an arbitrary contunuous function on~ . Let y (x) , y (x), y (x) be three solutions of the homogeneous 1 2 3 linear ODE + (sin x) y = 0 , Assume W(y , Y , Y i x) ~ 1 as x ~ - 00 123 Compute the Wronskian W(y , Y i x) . .., 1 3 [Hint: dont try to solve the Q 4. Find the power series expansion in x Cauchy problem ~ yll -x y = 0 i y(O) = 0, y/(O) = 1. of solution to the general solution of the Euler equation x 2 yll -x yl + y = 0 , x > 0 .
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F u')~ Y"' l r- \) .J,. 0. r- -+.i:> -=. a :.('~-'('- r + I -::.0 a.. -='(' - r-\\ -::. .') (r--\ )(r-I , / -lbj ~ 3<' ) ~~f\.ft.J. . .so\" ~ ~fI\()v-eY'-('a\M1 so \Y\ + ~M",,('" M. SQ\\t\ Or, ,~:r -tc. .::.o O\~\ I ~-=O, ~~~tb \= -.?:!: b~-(j«(. . - C> _ ~ Do. . !.I(~)
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Unformatted text preview: ~'\l.ll -:: .r;;;\'_ ----.. .. &quot;-~Q. ? l ) ~ 01~---= .;J. . j -+ / ~(())::.0 J J' t ()) =-1 v ~ ~ () (\ l)( -Xo ') V\ y\:&gt;C \A &lt;{ (\, ~ { )Y\ -\ tx J '- ::- (i\&quot; ()() ~ = ~\ Y\~I\ \~ ~II: ~l-Y\(Vl-I J 1 II ll\ _ ) 01\ XII-~-x [ '!-oo.A( II] &quot;'-&quot;., '(1&quot;''(\-'1-1. ~ U'O IV\ +z) (\\~\l GIn-\-,.l~ -~'K I -\-] L. ! 01&quot; (1&lt;)&quot;1 &quot;&quot; :::)~ l VI -\ i)( I,,\-l\ 0. &quot;n (1\.\&quot; -Y\ -::() I v0~\[)'~~~3\lxJ z: \ ~I(-\-) ~l-(-\-) ~j(~J' ~: (+) ~~'(-t) \A I (-D ~ Dz l c\-) . :J 3'( +) -'j s l-t)- ~ ; l-\: U . &quot;+ b\ t )\) 3' ( ) - ,,: l) '93 ( U + I I(' .~~( ) - ~ z () j: (t) . (5) 'j 1\ -X J z: o 'j:: ~ ().(\ (1\-Xo )(\ 'j :: t\&quot;:&gt;o . ) (0 :() 1'(0) =\ / .cD ' . ~ 10 + 3 y &lt;j + 3::J II + J = D...
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M 427K-Vishik-Spring 2003-Test 2 - ~'\l.ll -:: .r;;;\'_...

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