Solving Cauchy Problem Using Lagrange Method

Question

The question:

Solve the Cauchy Problem stated below using Lagrange Method:

$2Z_x-3Z_y+(x+y)z=0$

$z(x,0)=x^2 $

I attempted solving it by writing the characteristic equation:

$$\frac{dx}{2}=\frac{dy}{-3}=\frac{dz}{-(x+y)z} $$

Now, I figured I would need to determine the Lagrange multiplier $K_1, K_2, K_3$

Such that:

$$ \frac{dx}{2}=\frac{dy}{-3}=\frac{dz}{-(x+y)z}=\frac{K_1dx+K_2dy+K_3dz}{2K_1-3K_2-(x+y)zK_3}$$

Now, this was where I got stuck as I couldn't get the multipliers to solve further. Please help.

Answer 1

Was finally able to solve it. I guess it is all about mathematical manipulations.

The coefficient of $z$:

$(x+y) $

made me consider adding the numerator and denominator of the first two ratios in the characteristic equation. That is:

$K_1=1, K_2=1, K_3=0$

This gives:

$$\frac{dx}{2}=\frac{dy}{-3}=\frac{dz}{-(x+y)z}=\frac{dx+dy}{-1} $$

Then using the third and last ratio gives:

$$(x+y)(dx+dy)=\frac{dz}{z} $$

From here I can get solve for $z$