RAJALAKSHMI ENGINEERING COLLEGE ENGINEERING MATHEMATICS III MA1201 UNIT – I PARTIAL DIFFERENTIAL EQUATIONS PART - A 1) Find the partial differential equation of the family of spheres having their centres on the line x = y = z. 2) Obtain partial differential equation by eliminating arbitrary constants a and b from ( x − a) 2 + ( y − b) 2 + z 2 = 1. 3) Form the partial differential equation by eliminating a and b from z = ( x 2 + a 2 )( y 2 + b 2 ) . 4) Form the partial differential equation by eliminating the constants a and b from z = ax n + by n . 5) Find the partial differential equation of all planes passing through the origin. 6) Find the partial differential equation of all planes having equal intercepts on the x and y axis. xy 7) Eliminate the arbitrary function f from z = f ( ) and form the partial z differential equation. 8) Obtain partial differential equation by eliminating the arbitrary function from z = f ( x 2 + y 2 ). 9) Obtain the partial differential equation by eliminating the arbitrary functions f and g from z = f ( x + it ) + g ( x − it ). 10) Find the partial differential equation by eliminating the arbitrary function x 2 from φ[ z − xy, ] = 0. z ∂z ∂z 11) Find the complete integral of p + q = pq where p = and q = . ∂y ∂x 12) Write down the complete solution of z = px + qy + c 1 + p 2 + q 2 . 13) Find the singular solution of the partial differential equation z = px + qy + p 2 − q 2 . 14) Find the complete integral of the partial differential equation (1 − x) p + (2 − y )q = 3 − z. z x y = + + pq . 15) Find the complete integral of pq q p 16) Find the complete integral of p − y 2 = q + x 2 . 17) Find the solution of px 2 + qy 2 = z 2 . ∂3z ∂3z ∂3 z ∂3 z − 2 − 4 + 8 = 0. 18) Solve 3 ∂x ∂x 2 ∂y ∂x∂y 2 ∂y 3 19) Solve ( D 3 + D 2 D ′ − DD ′ 2 − D ′ 3 ) z = 0. 20) Find the particular integral of ( D 3 − 3D 2 D ′ − 4 DD ′ 2 + 12 D ′ 3 ) z = sin( x + 2 y ).
PART – B 1) Form the partial differential equation by eliminating the arbitrary functions f and g in z = f ( x 3 + 2 y ) + g ( x 3 − 2 y ). 2) Form the partial differential equation by eliminating the arbitrary functions f and g in z = x 2 f ( y ) + y 2 g ( x). 3) Form the partial differential equation by eliminating the arbitrary functions f and φ from z = f ( y ) + φ ( x + y + z ). 4) Find the singular solution of z = px + qy +
p 2 + q 2 + 16.
5) Solve z = px + qy + 1 + p 2 + q 2 . 6) Find the singular integral of the partial differential equation z = px + qy + p 2 − q 2 . 7) Solve z = 1 + p 2 + q 2 . 8) Solve p (1 − q 2 ) = q (1 − z ). 9) Solve 9( p 2 z + q 2 ) = 4. 10) Solve p (1 + q ) = qz. 11) Solve ( y − xz ) p + ( yz − x)q = ( x + y )( x − y ). 12) Solve ( y − z ) p − (2 x + y )q = 2 x + z. 13) Find the general solution of (3z − 4 y ) p + (4 x − 2 z )q = 2 y − 3x. 14) Solve x( y − z ) p + y ( z − x)q = z ( x − y ) 15) Solve y 2 p − xyq = x ( z − 2 y ). 16) Solve ( x 2 − yz ) p + ( y 2 − zx) q = z 2 − xy. 17) Solve ( x + y ) zp + ( x − y ) zq = x 2 + y 2 . 18) Solve ( x − 2 z ) p + (2 z − y )q = y − x. 19) Find the general solution of z ( x − y ) = px 2 − qy 2 . 20) Solve x( y 2 + z 2 ) p + y ( z 2 + x 2 )q = z ( y 2 − x 2 ). 21) Solve ( D 2 − DD ′ − 20 D ′ 2 ) z = e 5 x + y + sin(4 x − y ). 22) Solve ( D 2 + 4 DD ′ − 5 D ′ 2 ) z = 3e 2 x − y + sin( x − 2 y ). 23) Solve ( D 3 + D 2 D ′ − DD ′ 2 − D ′ 3 ) z = e 2 x + y + cos( x + y ). 24) Solve ( D 2 − DD ′ − 30 D ′ 2 ) z = xy + e 6 x + y . 25) Solve ( D 2 + DD ′ − 6 D ′ 2 ) z = x 2 y + e 3 x + y . 26) Solve
∂2z ∂2z ∂2z + − 2 = sinh( x + y ) + xy. ∂x 2 ∂x∂y ∂y 2
∂2z ∂2z ∂2z + − 6 = y cos x. ∂x 2 ∂x∂y ∂y 2 28) Solve ( D 2 − 5DD ′ + 6 D ′ 2 ) z = y sin x. 29) Solve ( D 2 + D ′ 2 + 2 DD ′ + 2 D + 2 D ′ + 1) z = e 2 x + y . 30) Solve ( D 2 − D ′ 2 − 3D + 3D ′) z = xy + 7. 27) Solve