Question

Let Q1=x(1.1) ,Q2=x(1.2), Q3=x(1.3). Then Let Q= ln(3 +|Q1|+ 2|Q2|+ 3|Q3|), Then T= 5 sin2(100Q)

1) where x=x(t) solves x′′+x= tan(t), x(0) = 1, x′(0) = 2

2) where x=x(t) solves x′′−x=te^t, x(0) = 1, x′(0) = 2.

3) where x=x(t) solves x′′−x=t^2, x(0) = 1, x′(0) = 2

4) where x=x(t) solves x′′−2x′+x=(e^t/2t), x(1) = 1, x′(1) = 2

Please show all steps and thank you!

Answer 1

Y "4X= tant , x100=1, x1(0)=2. " Auxiliary equation is given by mvf1zo * mati' is complementary functions Cifo Geost +Casimz Buy variation of find out particular parameter integral. we will J = cost, Je z simt, tatant Woonstion determinant Watant wal I 72 - 1 gel gel W I least losing sont cost a costat sinert 2 1 .

we know particularm integral PPS - Jos hy did + 0,13 -- core faime trointestet one ſecond epoca.com 2- cost sina tant, dit Sint I cost. tond treading Lx27 c-cost sinnt seet de tarwa ( simtdt 2 - cost sim't seet dat t + Sint sintat --cost fo-coitz)s er tdi sme femenz -cost 2)sectat at simt sint az =-cost seetat of cost costat et simt (cost) 2-cost log/seet& tant 1+ costsinza sinelost (1) en cost log (sectet tant l. Hence general Solofron is Duren Burg y cef & Polo Gcost & Casint – cost log (seet & tant,

vening given conditions, we get XCO)1 - 1- 9- 1x loglitol ef.921 X = 96-simt) of Ca cost - cost a (settentot sacz) + log Seedot tane simt Using (0)=2 (1) to 27 2= c2 -1x t 2= cq-1 C₂ e 3 hence net) a cost + 3 simt - cost log/sectitzant / auhich is the required value of x.

"--tet, X0=1) 1109= £. bay as Auxiliary equation is given mrizo af met -Coff complementos y Aumestrom) o get of ce et Pute ok, tet ( D +1 Rot 1 DA2D+1-1 72 st 2&t. ister ==== = [ [1+ $717

- $(-3+ -]+ se $ [taţ] 3 [ot] hence general soborone is given by Je Capit P.I [treating gex] & Xo Get ut eget teh hrt) using X(0), no la G+ C2 -0 XIC) gettazet (61)+ one (fington 4) + ( r 1 ditet Using x1(0)=2 I 2 = 9-C2 + 2y (orte) to 22 G-C2 - 4

- qn Cove+t- „ - © & oto 7 24 21+ = 42 792 12 13 + 2 = 1 hence X(4)= 4yet Ete (19). cohreh in the requiered walue of xo

3) x_x= tv, x(0)21, X1109=2. Auxiliary equation is given by mkizo 29 m afl is caps get thet P.R= Len DEI antoren 2 [1-Dw] len u a. [1+ Dr Dort and I tr .. [++ 2 ] General soil is given by Je cop & P.I L treating yax] 27 = geta ezēt – (24f")

Using Condition tool. a la atca-2 946=3. - X(t) = Goto Creta et using X1 (6)=2 no es que2 - no 0+0=) 2455 * = 4 e) 5GF2 as 3-2 > zed Hence Alt) Bett té t – (2+2"). which is the required value

Here, * XI!_2014x-1 ), x(i)=1, X(X)=2. Auxiliary equation is givens wory mkem.+1=0 ay ( ) meln. "Cipa Gregtet = 24. Dat langs die et - berita et to logt De Slegtdt. & General solution is X=J= (Gtcat et lest de .

using X(1)= 1 1=16+C)e + o - G4C2 = te o X117)- (1409 )ett et (09) + b/1ggede xet ther.stat -7%!= (GtCot)et + etice + b/legedz tez ed et logt usimy x(1)=2 an 22 (4+C2) et eget of e GetCa (ete) = 2 et Get 2222 =2 of 4+2 C2 = 2. -*-=0224-heste un 27 + 2te & 920.

Hena *(+)-betet + $t (1934at. which is the required value of a thence the result.