1. (b) First order. y: 2sinxcosx #9 sin2x so No.

No. 2: 2(3sinx)X3apx) does

= 2sin2x (because 2sinxCDX = sin 2x) so so No. (h) yi+2xy,-1=-2xAe xe dt + -xx Aex ex²+2 3: 2(ex) (ex) #9 sin2x -x² +2Axe Sedt-1=0 only

起

if A=1. Thus, in general, No. 2+2x-1=-2xe a ex all choices of a, so Yea. + 2xe Sa etat edt -1 does = 0 for

xex et² dt + ex

3. 43z, we obtain Evaluating Mxx, Ugy, 4zz, Uxx + Myy 4 Mzz = (c²-a²-b²) sinax sin by sinh cz = 0 if c²= a²+b² (or if a=b=c =0 so that sinax sinby sink.cz =0, but this is a sub case of c²= a²+b²)!

5. (b) y'+3y² = xexx + 382x = e^x (2+3e^x). The ex factor is not o for any x, let alme for all x. And for the second factor to be O for all x requires that exx is a constant and that, in turn, reguris that 2=0. But if 1=0 then 1+3exx =0+3+0. Thus, no such solutions.

(c) M3m²+2y=(x-3)+2)=0 if x²-3x+2=0, ie, if x = 102 2. Thus, are

6. (b) y"-y-x² = (-2+Asinkx+Bashx)-(-x-2+ Asanhx+Bashx)-x² does=0. (0)=-2=-2+B and y'(0)=0= A give A=B=0, so y(x) = -x²-2、 7. (b) Nonlinear due to the sy tem

yy (d) Nonlinear due to the exp(y) term (9) Nonlinear due to the yoy" "tim.

8. "y" C, since yo² <<1>

All others linear.

Section 1.3