Question

Use the change of variables to solve the differential equation. 5. xy' = y +2Vxy dy Y-3 dx y + x 7. cy' + y ln x = y lny 8. (x + yey/t) dx – xey/dy = 0, y(1) = 0)

Answer 1

© ny' = y + 2 vay yet, y=04. then, y'= ut av So from cili utaw=0+ 210 is, et ont = x On integrating: Select = S dhe +A c API.c] ie, VF = 4+A in V = x +A i, vg = 2* Tara o dy = ) ... (1) Let Y=vx CS Carascanud aut XV ? Scanned with

se from (ili verw = la LÀ, xv - - -v - - -v ) is to the war was conduce a pohod) dw = x de On integrating : Si laut + show) dx = fue de + Ay [A=IcJ ie, I in litery + tant (u) = x +A ie, in (H 4") +22am* (W) = atta i, m (1+ y ) + 2 land (Y) = x +A Byl = y m( 3 ). ...) Let y = ux i, y' = 0+ 2.0 So, from (i): utaw: Un cu). is, we = x-1 uclmcwi) ie, du 3 Scann uamu-1) CamScanner

On integrating : (dlmcws) - se ole + Inceste= I.c] :) Thews- je, in (hn (0)-1) = ln (X) + m cc). ie, ln (m (0)-1) = ln Cen). i, in (W)-1=CK ie, in (W) = lten ů, uzelten i, y=x. eltex ® aty e *) dhe-xe dy zo Let, y=w* in, yl= utaw So from (i): (u + xwe). eu = 1+ Wied ie, u.ee te we U = Itupu i, e".*.= ţ cs scamiler ne dv=de CamScanner

Un integrating : Seudu = som t t mc [e = I.c] i, eu = ln (CX) - - w, e amcom) --- 031 Putting y(x) = 0 in (i): i= m ces ü, ere So, foram (ii), a = m (en) CS Scanned with CamScanner