1. '3
1.(D 3 -7DD ' -6DD )Z=sin(x+2y)+e 2 x y
1 1 (x 2 y)
ANS: 1 (y x) 2 (y 2 x) ( y 3x) + [cos(x 2 y)] + [e ]
3
75 12
'2
2. .(D 2 +3DD ' -4D )Z=siny+x
x3 1
ANS: 1(y 4 x) 2(y x) + + sin y
2 4
2 ' '2
3. .(D +DD -6D )Z=x 2 y+ e 3 x y
x4 y x5
ANS: 1(y 2 x) 2(y 3 x) + -
12 60
4.Solve:px+qy=z,Using lagrange’s multipliers.
x y
ANS: ( , ) 0
y z
5. Solve:pzx+qzy=xy,Using lagrange’s multipliers.
x x
ANS: ( , ( ) y 2 z2) 0
y z
6. Solve:x(y-z)p+q(z-x)=Z(x-y),Using lagrange’s multipliers.
ANS: f ( x y z, xyz) 0
7. . Solve:(mz-ny)p+q(nx-lz)q=ly-mx Using lagrange’s multipliers.
ANS: f ( x 2 y2 z 2 , lx my mz) 0.
8.Eliminate f from z=f(x 2 +y 2 )
p x
ANS:
q y
9.Eliminate a & b from z=ax+by+a 2 +b 2
ANS:z=px+qy+p 2 +q 2 .

2. 10. Eliminate a & b from z=(x+a 2 )(y+b 2 )
ANS:4xyz=pq.
11.Find the Partial differential equation of all planes through the origin.
ANS:px+qy-z=0