Solve the following Lagrange’s Partial Differential Equations : (yxp –xzq = xy

 Solve the following Lagrange’s Partial Differential Equations :

(yxp –xzq = xy

Solution :To solve a partial differential equation (PDE), we need to first identify the dependent variables, the independent variables, and the order of the PDE. In this case, the dependent variables appear to be y and z, while the independent variables appear to be x and p. The order of the PDE can be determined by counting the highest order of differentiation present in the equation. In this case, the highest order of differentiation is first order, so the PDE is a first-order PDE. To solve this PDE using the method of Lagrange's, we need to rewrite the equation in standard form. To do this, we can rearrange the terms to get: xy - yxp + xzq = 0

Next, we need to find a function F(x,y,z,p) such that:

Fx - yFp + xFq = 0

Comparing this to the equation we started with, we can see that if we let F(x,y,z,p) = xy - yxp + xzq, then the equation is in standard form.

To solve for y and z, we can use the method of characteristics. The characteristics are the curves in the x-y plane along which the solution to the PDE is constant. These curves can be found by solving the following system of equations:

dx/x = dy/y = dz/z = dp/q

Solving this system of equations, we find that the characteristics are given by:

x^2 - y^2 = c

where c is an arbitrary constant.

To find the general solution, we can set up an equation for each characteristic curve and solve for y in terms of x. For example, for the characteristic curve x^2 - y^2 = c1, we can solve for y to get:

y = sqrt(x^2 - c1)

Substituting this expression for y back into the equation F(x,y,z,p) = 0, we can solve for z in terms of x. For example, if we choose c1 = 1, then we can solve for z to get:

z = sqrt(x^2 - 1)

This gives us the general solution to the PDE in parametric form:

y = sqrt(x^2 - c1) z = sqrt(x^2 - 1)

where c1 is an arbitrary constant.

This is the general solution to the PDE. To find a particular solution, we need to specify the values of y and z at some point (x0,y0,z0). We can then use the general solution to determine the values of the arbitrary constants c1, c2, etc. that will give us the desired solution.