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Mah 0 Homwork S 6 Soluions 0 oins. ( ps I ll lav i o you vrify ha h omplimnary soluion is : y ( os( sin ( Th guss for h pariular soluion and is drivaivs ar, +. ( os( sin ( ( os( ( sin ( Y ( D 6B os( + ( B 6D sin ( Y A + B + D Y B + D + D B No ha w nd o add in a o h sin/osin rm baus hy ar h omplimnary soluion. lugging his ino h diffrnial quaion, simplifying and olling lik rms givs, ( ( + ( + ( D os Bsin 6A sin 9 os Sing offiins qual and solving givs, os( : D A sin ( : B B : 6A 9 D Th gnral soluion is hn, 0 ( os( sin ( os( sin ( y + + 9 6. ( ps I ll lav i o you vrify ha : ( os( y + sin ( pariular soluion is, 9 6. Th guss for h ( sin ( ( os( Y A + B + C + D + E + F + G No h on h sond and hird rms baus a porion of hs rms (D and G is h omplmnary soluion. y +. Now,. ( ps I ll lav i o you vrify ha h omplimnary soluion is : ( 0 ( 5 W g 0 Don forg o divid by h o g a in fron of h sond drivaiv rm. 5 0 5 0 ( ( ( Y d + d 5 0 5 0 d d + Th gnral soluion is hn, 6 0 ( 6 y + + + Diffrniaing and applying h iniial ondiions givs, + + 0 5 9 6 9 9 6 6 y + + + + + 9 9 0 5 9 0 Th aual soluion is hn : (

Mah 0 Homwork S 6 Soluions 0 oins. ( ps Hr ar h ky quaniis for h problm. Th IV is, m L k 6 ω 6 6 0 ( ( u + 6u 0 u 0 u 0 Th gnral soluion is, u Applying h iniial ondiions givs, Now, rdu down o a singl osin. ( os( + sin ( ( os( sin ( u ( ( ( R + 0.59 δ an 0.9 δ δ + π.966 In his as h firs angl is h orr on and so h singl osin form of h soluion is, ( 0.59 os( + 0.9 u No Gradd. I ll lav i o you vrify ha h omplimnary soluion is : ( y +. Th guss for h pariular soluion and is drivaivs ar, Y A + B + C + D Y A + B + C Y 9A + B lugging his ino h diffrnial quaion, simplifying and olling lik rms givs, 5 A B C + B D + Sing offiins qual and solving givs, : 5A A 5 : B B : C 0 C 0 0 : B D D Th gnral soluion is hn, ( y + + 5. I ll lav i o you vrify ha h omplimnary soluion is : ( 6 y +. Th guss for h pariular soluion and is drivaivs ar, Y A + B Y A + ( B + B Y 6A + ( 6B + 9B No h on h sond rm o ak ino aoun h fa ha i was in h pariular soluion. lugging his ino h diffrnial quaion, simplifying and olling lik rms givs,

Mah 0 Homwork S 6 Soluions 0 oins 0A B 5 Sing offiins qual and solving givs, : 0A 5 A : B B Th gnral soluion is hn, 6 ( + + + y Taking h drivaiv and applying h iniial ondiions givs, + + + 6 + Th aual soluion is hn, 6 9 9 6 ( + y 6 y +. Th guss for 5. I ll lav i o you vrify ha h omplimnary soluion is : ( 0 0 h pariular soluion is, 0 0 Y A + B + Dos + Esin + F + G + H ( No h on h firs rm. I aks boh s o g all porions of ha ou of h omplimnary soluion. 6. I ll lav i o you vrify ha h omplimnary soluion is : y ( os( + sin (. Th informaion w nd for Variaion of aramrs. os( sin ( W g( os sin sin + os ( ( ( os + sin Th pariular soluion is hn, sin ( ( os( ( Y os( d + sin ( d os sin d sin os d Th gnral soluion is hn, ( ( ( ( ( ( os sin ( os( + sin ( y

Mah 0 Homwork S 6 Soluions 0 oins 9. Hr ar h ky quaniis for h problm. ( 0. 9. 0.05 ( m 0. L 0.05 k 56. γ γ CR 56. 0.. Th IV is, ( ( 0.u +.u + 56.u 0 u 0 0.5 u 0 0.0 Th gnral soluion is (I ll lav i o you o vrify, Applying h iniial ondiions givs, ( + u ( 0.5. u 0. Taking wha w an from # w hav h following, ( ( u + 6u 0 u0 u 0 u os + sin No ha baus ω ω0 w WILL hav rsonan. Undrmind Coffiins will probably b h asis for a pariular soluion so, U Aos + Bsin Bos Asin sin ( ( ( Sing offiins qual and solving givs A, B 0. Th gnral soluion is hn, u os + sin sin ( Applying h iniial ondiions givs, u os + sin sin Now, rdu down o a singl osin. ( 96 ( ( ( 96 R + 96 0.9 δ an 0.0 δ δ+ π. In his as h firs angl is h orr on and so h singl osin form of h soluion is, u 0.9 os 0.0 sin. Taking wha w an from #9 w hav h following, 0.u +.u + 56.u 0 u 0 0.5 u 0 0.0 u + Undrmind Coffiins will probably b h asis for a pariular soluion so, U Aos + Bsin ( ( 6 6 ( A+ B os( + ( B A sin ( sin ( os( 5 5 5 5 Sing offiins qual and solving givs A 0.0095, B 0.005. Th gnral soluion is hn, u( + 0.0095os( + 0.005sin ( Applying h iniial ondiions givs, u 0.05.05 0.0095os + 0.005sin

Mah 0 Homwork S 6 Soluions 0 oins Now, rdu down o a singl osin. 0.005 R 0.0095 + 0.005 0.09 δ an 0. δ δ + π. 0.0095 In his as h sond angl is h orr on and so h soluion is hn, ( 0.05.05 + 0.09os(. u

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