Mathematical Physics Vol 1

Chapter 7. Partial differential equations

382

7.8 Examples

Problem 245 Find the general solution of equation

∂ f ∂ x (= f x )= 0 , where f = f ( x , y ) .

Solution

∂ f ∂ x

= 0 ⇒ f = f ( y ) , where f is an arbitrary differentiable function of the variable y .

Problem 246 Find the general solution of equation 14 ∂ 2 f ∂ x ∂ y

= 0 .

Solution

∂ x or ∂ ∂ y

∂ f ∂ y ∂ f ∂ x

    ∂

= 0 ,

⇒

= 0 ,

 

∂ f ∂ y or ∂ f ∂ x

= ϕ ( y ) ⇒ f = R ϕ ( y ) d y + ψ ( x ) , = φ ( x ) ⇒ f = R φ ( x ) d x + γ ( y ) .

⇒

 

Thus, the general solution is a function of the form f = ϕ ( x )+ ψ ( y ) , where ϕ and ψ are arbitrary differentiable functions of x and y , respectively.

14 The given equation is a partial second order equation. However, by a simple substitution ( ∂ f / ∂ y = ϕ ( y ) or ∂ f / ∂ x = φ ( x ) ) it is reduced to a first order equation, which is why it is placed here.