Question

Define a partial differential equation and give an example.

Hence find the differential equation arising from F[y/x,z/x³]=0

Answer 1

The partial differential equation is an equation which imposes relations between the various partial derivatives of a multivariable function. For exaple, $\frac{d^2u}{d x^2}-\frac{d^2u}{d y^2}=0.$

Let's find the differential equation arising from $F[y/x,z/x3]=0$ .

Let $f(x;y;z)=F\left(\frac{y}{x};\frac{z}{x^3}\right).$

$f'(x)=-\frac{y}{x^2}F'_{x_1}-3\frac{z}{x^4}F'_{x_1}; f'(y)=F'_{x_1}/x; f'(z)=F'_{x_1}/x^3.$ So we have $F'_{x_1}=xf'(x); F'_{x_2}=x^3f'(z)$ . Since the function $x(y;z)$ can be given by from the equality $f(x;y;z)=0$ by the implicit function theorem. We have $x'(y)=-\frac{f'(y)}{f'(x)}, x'(z)=-\frac{f'(z)}{f'(x)}.$ From the equality $f'(x)=-\frac{y}{x^2}F'_{x_1}-3\frac{z}{x^4}F'_{x_1}$ we have $1=\frac{y}{x}x'(y)+3\frac{z}{x}x'(z)$ or $yx'(y)+3zx'(z)=x.$