Mathematical Physics Vol 1

Chapter 7. Partial differential equations

336

Example 233 Let f 1 ( ξ ) and f 2 ( η ) be arbitrary differentiable functions, where ξ = y + ax , and η = y − ax . Determine the partial differential equation, whose solution satisfies the equation z = f 1 ( ξ )+ f 2 ( η ) .

Solution Using the Monge notation, we obtain

∂ f 1 ∂ x ∂ f 1 ∂ y

∂ f 2 ∂ x ∂ f 2 ∂ y

∂ f 1 ∂ξ ∂ f 1 ∂ξ

∂ξ ∂ x ∂ξ ∂ y

∂ f 2 ∂η ∂ f 2 ∂η

∂η ∂ x ∂η ∂ y

∂ z ∂ x ∂ z ∂ y

d f 1 d ξ −

d f 2 d η

= a f ′ 1 − f ′ 2 ,

p =

= a

a

=

+

=

+

d f 1 d ξ

d f 2 d η

= f ′ 1 + f ′ 2 ,

q =

=

+

=

+

=

+

( p )= a

∂ f ′ 2 ∂ x

= a

∂η ∂ x

∂ 2 z ∂ x 2

∂ ∂ x

∂ f ′ 1 ∂ x −

∂ξ ∂ x −

d f ′ 1 d ξ

d f ′ 2 d η

r =

=

=

= a 2 f ′′

′′ 2 ,

1 + f

(7.5)

∂ 2 z ∂ y 2

∂ ∂ y

∂ f ′ 1 ∂ y

∂ f ′ 2 ∂ y

∂ξ ∂ y

∂η ∂ y

d f ′ 1 d ξ

d f ′ 2 d η

t =

( q )=

=

+

=

+

=

= f ′′

′′ 2 .

1 + f

(7.6)

From (7.5) and (7.6) we obtain the required partial differential equation r − a 2 t = 0 .

Example 234 Let a function be given in the form

z = a 2 x 2 + 2 abxy + b 2 y 2 + cx + dy + e ,

where a , b , c , d and e are arbitrary constants. Write the differential equation this function satisfies.

Solution