Download Assignment 3 posted on 19/01/2016

Mathematics (MA-002)
Assignment 3: Partial Differential Equations
Spring Semester 2015-16
1. Find the partial differential equations by eliminating the arbitrary constants from the following
equations:
√
(a) z = ax + y x2 − a2 + b
(c) 2z =
2
x
a2
+
2
y
b2
(b) z = axey + 12 a2 e2y + b
(d) log(az − 1) = x + ay + b
2. (a) Find the partial differential equation of all planes which are at a constant distance from
origin.
(b) Find the partial differential equation of the set of all right circular cones whose axes coincide
with z-axis.
(c) Find the partial differential equation of all cones which have their vertex at the origin.
(d) Find the partial differential equation of the family of planes, the sum of whose x,y,z intercepts is
equal to unity.
3. Find the partial differential equation by eliminating arbitrary functions from the following
relations
(a) z = f (xy) + g( xy )
y−b
(d) f ( x−a
z−c , z−c ) = 0
(b) f (x + y + z, x2 + y 2 + z 2 ) = 0
(c) lx + my + nz = φ(x2 + y 2 + z 2 )
(e) f (x + y + z) = xyz
4. Find the general solution of the following partial differential equations
(a) cos(x + y)p + sin(x + y)q = z
(d) z(xy + z 2 )(px − qy) = x4
(b) py + qx = xyz 2 (x2 − y 2 )
(c) (x2 + 2y 2 )p − xyq = xz,
(e) (x2 − y 2 − yz)p + (x2 − y 2 − zx)q = z(x − y)
5. Find the general solution of the following Lagrange linear equations
(a) (y + z)p + (z + x)q = x + y, (b) (x2 − yz)p + (y 2 − zx)q = z 2 − xy (c) px + qz + y = 0
p
(d) (p − q)z = z 2 + (x + y)2
(e) px + qy = z − a (x2 + y 2 + z 2 )
6. Solve the following partial differential equations ;
(a) p + q + pq = 0, (b)
p2 + 6p + 2q + 4 = 0 (c) p2 + q 2 = npq (d) p = eq ,
(e) p2 + q 2 = m2 .
7. Solve the following partial differential equations ;
p
p
(a) z = px + qy + 4 1 + p2 + q 2 , (b) z = px + qy + 1 + αp2 + βq 2
(c) pm sec2m x + z l q n coesc2n y = z lm/(m−n) (d) z = x2 p2 + y 2
√
(e) z = px + qy + 2 pq
(8) Solve the following partial differential equations
(a) p(1 − q 2 ) = q(1 − z) (b) z 2 (p2 + q 2 + 1) = 1 (c) 4(1 + z 3 ) = 9z 4 pq
(d) p(1 + q) = qz
(e) p2 = z 2 (1 − pq)
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(9) Solve the following partial differential equations
(a) px + q = p2 (b) p2 y(1 + x2 ) = qx2
(d) q = px + p
2
(c) q(p − cosx) = cosy
(e) x(1 + y)p = y(1 + x)q
Answers
1. (a)
∂z
= a ∂y
+ xy
∂z ∂z
∂x ∂y
2. (a) z = px + qy + a
(b)
∂z
∂y
∂z
∂z 2
= x ∂x
+ ( ∂x
) (c) 2z = px + qy (d) (1 +
p
p2 + q 2 + 1,
∂z
∂z
(b) y ∂x
= x ∂y
,
∂z ∂z
∂y ) ∂x
∂z
= z ∂y
(c) px + qy = z (d) px + qy − z =
pq
p+q−pq
∂z
3. (a) x2 r − y 2 t + xp − yq (b) p(y + z) − (x + z)q = x − y (c) (nx − mz) ∂x
+ (lz − nx) ∂z
∂y = mx − ly
(d) (x − a)p + (y − b)q = z − c (e) x(y − z)p + y(z − x)q = z(x − y)
4. (a) φ[z
√
2
(c) f (yz, x2 y 2 + y 4 ) = 0
(d) φ(xy, x4 − z 4 − 2xyz 2 ) = 0
2
5. (a) f ( x−y
y−z , (x − y) (x + y + z)) = 0
(d) f (x+y,
2
π
y−x
cot( x+y
(sin(x + y) + cos(x + y))] = 0
2 + 8 ), e
1
2
(b) f (x2 − y 2 , y2 +
log z 2 + (x + y)2 )=0 (e) φ( xy , √
x
)
x2 +y 2 +z 2
2
=0
)=0
(e) φ(z − x + y, x z−y
2
(b) f (xy + yz + zx, x−y
y−z ) = 0
a−1
2
1
z(x2 −y 2 ) )
(c) f (y 2 + z 2 ,
x
etan
−1 ( y )
z
)=0
=0
2
+ ψ(a)
(b) z − ax + ( a2 + 3a + 2)y − ψ(a) = 0
√
(c) z − ax − a2 (n + n2 − 4)y − ψ(a) = 0
(d) z − ax − log(ay) − ψ(a) = 0
p
(e) z = ax + y (m2 − a2 ) + φ(a).
6. (a) z = ax −
a
1+a y
p
7. (a) z − ax − ψ(a)y − 4 1 + a2 + (ψ(a))2 = 0
(c)
m−n−l
m−n
m−n
m−n−l z
√
1−a2
a
(d) x y
= a4 (2x + sin2x) +
= ψ(a)e
√
2 z
1
(1−am ) n
4
(b) z − ax − ψ(a)y −
p
αa2 + β(ψ(a))2 + 1 = 0
(2y − sin2y) + ψ(a)
p
(e) z − ax − ψ(a)y − 2 (a)ψ(a) = 0
8. (a) 4(1 − a + az) − (x + ay + φ(a))2 = 0 (b) (1 + a2 )(1 − z 2 ) − (x + ay + φ(a))2 = 0
(c) a(1 + z 3 ) − (x + ay + φ(a))2 = 0
(d) az − 1 = φ(a)ex+ay
√
1
(e) ±(x + ay + φ(a)) = − sinh−1 ( z√
) + 1 + az 2
a
√
√
9. (a) z = 41 (x2 + x x2 + 4a) + a log(x + x x2 + 4a) + ay + ψ(a)
√ √
(b) 2z = 2 a 1 + x2 + ay 2 + b = 0
(c) z − ax − sinx − a1 siny + b = 0
√
√
2
(d) z − x4 ± 21 ( x2 x2 + 4a + 2a log(x + x2 + 4a)) + ay + φ(a) = 0
(e) z = a(log xy + (x + y)) + φ(a).
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