Find the canonical form of the following PDE:
u x x − 6 u x y + 9 u y y = x y 2 \boldsymbol {u} _ {x x} - 6 \boldsymbol {u} _ {x y} + 9 \boldsymbol {u} _ {y y} = x y ^ {2} u xx − 6 u x y + 9 u yy = x y 2
**Be sure to show the change of coordinates that reduces the PDE to canonical form.**
**Solution:**
a = 1 , b = − 3 , c = 9 a = 1, b = - 3, c = 9 a = 1 , b = − 3 , c = 9 Δ = b 2 − a c = ( − 3 ) 2 − 1 ⋅ 9 = 0 , so the equation is parabolic. \Delta = b ^ {2} - a c = (- 3) ^ {2} - 1 \cdot 9 = 0, \text{ so the equation is parabolic.} Δ = b 2 − a c = ( − 3 ) 2 − 1 ⋅ 9 = 0 , so the equation is parabolic.
Write down the characteristic equation:
d y 2 + 6 d x d y + 9 d x 2 = 0 d y ^ {2} + 6 d x d y + 9 d x ^ {2} = 0 d y 2 + 6 d x d y + 9 d x 2 = 0 d y d x = − 3 \frac {d y}{d x} = - 3 d x d y = − 3 y = − 3 x + C y = - 3 x + C y = − 3 x + C
Determine new variables ξ \xi ξ η \eta η
ξ = φ ( x , y ) = 3 x + y \xi = \varphi (x, y) = 3 x + y ξ = φ ( x , y ) = 3 x + y φ x = 3 , φ y = 1 , φ x x = 0 , φ x y = 0 , φ y y = 0 \varphi_ {x} = 3, \varphi_ {y} = 1, \varphi_ {x x} = 0, \varphi_ {x y} = 0, \varphi_ {y y} = 0 φ x = 3 , φ y = 1 , φ xx = 0 , φ x y = 0 , φ yy = 0 η = ψ ( x , y ) = x \eta = \psi (x, y) = x η = ψ ( x , y ) = x ψ x = 1 , ψ y = 0 , ψ x x = 0 , ψ x y = 0 , ψ y y = 0 \psi_ {x} = 1, \psi_ {y} = 0, \psi_ {x x} = 0, \psi_ {x y} = 0, \psi_ {y y} = 0 ψ x = 1 , ψ y = 0 , ψ xx = 0 , ψ x y = 0 , ψ yy = 0 ∣ φ x ψ x φ y ψ y ∣ = ∣ 3 1 1 0 ∣ ≠ 0 \left| \begin{array}{c c} \varphi_ {x} & \psi_ {x} \\ \varphi_ {y} & \psi_ {y} \end{array} \right| = \left| \begin{array}{c c} 3 & 1 \\ 1 & 0 \end{array} \right| \neq 0 ∣ ∣ φ x φ y ψ x ψ y ∣ ∣ = ∣ ∣ 3 1 1 0 ∣ ∣ = 0 u x x = u ξ ξ φ x 2 + 2 u ξ η φ x ψ x + u η η ψ x 2 + u ξ φ x x + u η ψ x x = 9 u ξ ξ + 6 u ξ η + u η η u _ {x x} = u _ {\xi \xi} \varphi_ {x} ^ {2} + 2 u _ {\xi \eta} \varphi_ {x} \psi_ {x} + u _ {\eta \eta} \psi_ {x} ^ {2} + u _ {\xi} \varphi_ {x x} + u _ {\eta} \psi_ {x x} = 9 u _ {\xi \xi} + 6 u _ {\xi \eta} + u _ {\eta \eta} u xx = u ξξ φ x 2 + 2 u ξ η φ x ψ x + u ηη ψ x 2 + u ξ φ xx + u η ψ xx = 9 u ξξ + 6 u ξ η + u ηη u x y = u ξ ξ φ x φ y + u ξ η ( φ x ψ y + φ y ψ x ) + u η η ψ x ψ y + u ξ φ x y + u η ψ x y = 3 u ξ ξ + u ξ η u _ {x y} = u _ {\xi \xi} \varphi_ {x} \varphi_ {y} + u _ {\xi \eta} \left(\varphi_ {x} \psi_ {y} + \varphi_ {y} \psi_ {x}\right) + u _ {\eta \eta} \psi_ {x} \psi_ {y} + u _ {\xi} \varphi_ {x y} + u _ {\eta} \psi_ {x y} = 3 u _ {\xi \xi} + u _ {\xi \eta} u x y = u ξξ φ x φ y + u ξ η ( φ x ψ y + φ y ψ x ) + u ηη ψ x ψ y + u ξ φ x y + u η ψ x y = 3 u ξξ + u ξ η u y y = u ξ ξ φ y 2 + 2 u ξ η φ y ψ y + u η η ψ y 2 + u ξ φ y y + u η ψ y y = u ξ ξ u _ {y y} = u _ {\xi \xi} \varphi_ {y} ^ {2} + 2 u _ {\xi \eta} \varphi_ {y} \psi_ {y} + u _ {\eta \eta} \psi_ {y} ^ {2} + u _ {\xi} \varphi_ {y y} + u _ {\eta} \psi_ {y y} = u _ {\xi \xi} u yy = u ξξ φ y 2 + 2 u ξ η φ y ψ y + u ηη ψ y 2 + u ξ φ yy + u η ψ yy = u ξξ 9 u ξ ξ + 6 u ξ η + u η η − 6 ( 3 u ξ ξ + u ξ η ) + 9 u ξ ξ = η ( ξ − 3 η ) 2 9 u _ {\xi \xi} + 6 u _ {\xi \eta} + u _ {\eta \eta} - 6 \left(3 u _ {\xi \xi} + u _ {\xi \eta}\right) + 9 u _ {\xi \xi} = \eta (\xi - 3 \eta) ^ {2} 9 u ξξ + 6 u ξ η + u ηη − 6 ( 3 u ξξ + u ξ η ) + 9 u ξξ = η ( ξ − 3 η ) 2
The canonical form is:
u η η = η ( ξ − 3 η ) 2 u _ {\eta \eta} = \eta (\xi - 3 \eta) ^ {2} u ηη = η ( ξ − 3 η ) 2 
#340153 on Dec 2023