Question

3:43 LTE Х P204.SU2O_MT1_7-... MAT(40pt pts vachtel ASSUME the Differential Equation Alsomethi Showhow/whry the following trigonometric expression Sow IS NOToolutie does the consists the differential equation 2 (why/howar why how so? Il Show how/by the following expertial expressive ISIS NOT solution does the expression the differential equation 2 (why/how or why/how not?) L Show how/boy the following trigonometric exprensic Sc SNOT HARMONIC OSCILLATOR Woes the consent the diletilegt for harmonic 2 , who determine is PERIOD and expressit in terms oli.comstants Show how/weby the following trigonometric exprice ISISNOT. HARMONIC OSCILLATOR | Woes the coast the differentialematice for barn 2 , why wo determine is PERIOD and expresik in terme en constants) Page 3 of 8 ASSUME the Duterial

Answer 1

a) x R3 it x= xocos xko dt at R3 2 x kg R3 - Ho coskot+TI kg)++ dt? Lus= d²x 2 -Xo cos . (ko (i) dx df ² To + Th 1 is a I love you prove solution of O or NOT differenticete the expresion : dze - dbo.cos (49) + + ) =-26 sin(kg t. + T/ to sinko d² d Ho sinfuglt dt at + TTA Iz luke substituting da &a in o: ([49]+ 9.74 +T%) 44 ata RHS - (hot) x = - ko x to co ko ) + + 7 -**4+%) * LUS & RUS ginu brigrometric, expresion is NOT a solution J the differential 1. cosine expression does not satisfy the differential equation, ilhamento 10) ilkolo xo e teksje xC iko k3 f + TVA = x kg R3 2. Similarly as above As above. X = Hoe da co e dt d² It² X i klo R: xo e R3)*x/kg substituting site : - No e'l R3 (kg Х 코 t² K3 f Lus = d²x RUS R3 LUS € RUS (ko) a ke. R3 ro R 1. The exponential expression ion does not satiks the differential equation elementet 2. Byl Calenlating dn 4 substituting in giren equation prores not the solution x= to e ata is

iii). WE COS a t R3 : velocity dă dt Consider equation of displacement ci sin ( centa detergente alt & The K] xka K3 Now acceleration a= -i cos 2 t R3 is a constant CC - lufte u R] :a= C i cos R² a L 2x it + 1 is a 2T in 1) The alt + T1/2 =(xx which implies force Fax force Fax & Hence x = i cost i online This Harmonic Oscillator of frequency - angular frequency (A and Hence Time posiad - L zafkort =T frequeny 278 expression setúlies the harmonie mscillator differential equation as "force 2 displacement. in Similarly x = i šin ( )*+18] dx sicos [ (Kejt + 7/7 x ke dx (43)* + tai? Kereta a displacement 1) Force a displacement - expression is a harmonic oscillator 2) Time period =T= frequency 2) Period - T - 27 ke K3 as above.. ke. It -isin afz acceleration Given 2TR² 28 KQ R3x27 КО

Then the method remains the same, derivate the given expression to get d2x/dt2 and subtitute in the given differential equation.

As LHS = RHS, i.e. if the expression satisfies the differential equation then the expression is a solution of the harmonic oscillator. If the expression does not satisfy then it is not a solution.