Question

For the equation x^2y" - xy' = 0, find two solutions, show that they are linearly independent and find the general solution. Equations of the form ax^2y" + bxy' + cy = 0 are called Euler's equations or Cauchy-Euler equations. They are solved by trying y = x^r and solving for r (assume that x greaterthanorequalto 0 for simplicity).

Answer 1

Answer:

$y=A+Bx^{2}$

Step-by-step explanation:

The given Cauchy-Euler equation is: $x^2y''-xy'=0$

Comparing to the general form: $ax^2y''+bxy'+cy=0$, we have a=1,b=-1 and c=0

The auxiliary solution is given by: $am(m-1)+bm+c=0$

$\implies m(m-1)-m=0$

$\implies m(m-1-1)=0$

$\implies m(m-2)=0$

$\implies m=0\:\:or\:\:m=2$

The general solution to this is of the form $y=Ax^{m_1}+Bx^{m_2}$, where A and B are constants.

$y=Ax^{0}+Bx^{2}$

Therefore the general solution is;

$y=A+Bx^{2}$

Let $y_1=A$ and $y_2=Bx^2$

Since we CANNOT express the two solutions as constant multiple of each other, we say the two solutions are linearly independent.

$y_1\neCy_2$, where C is a constant.