Answer to Question #104562 in Differential Equations for NILESH SAINI

Question

iii) The p.d.e.
auxx + 2b uxy + c uyy = 0
where a, b ,c are constants is irreducible when
b 2− ac =0 .

Answer 1

Let's rewrite the equation in operational form

$F(D,D')=aD^2+2bDD'+cD'^2=0$

An equation is irreductable if it can't be written as a product of linear factors, hence

$F(D;D')=(a_1D+b_1D'+c_1)(a_2D+b_2D'+c_2)=a_1a_2D^2+$

$+(a_1b_2+a_2b_1)DD'+b_1b_2D'^2+(a_1c_2+c_1a_2)D+(b_1c_2+c_1b_2)D'+c_1c_2$

Using the method of undetermined coefficients, we assume

$a_1a_2=a; b_1b_2=c; a_1b_2+a_2b_1=2b;$

$b_1c_2+c_1b_2=a_1c_2+c_1a_2=c_1c_2=0$

Hence $c_1=c_2=0$

If $b^2-ac=0,$ $a_1=a_2=\alpha; b_1=b_2=\beta$ ,therefore $F(D,D')=(\alpha D+\beta D')^2$ which is irreducible