Transform the following problem into standard form Uxx yUyy

Transform the following problem into standard form:

U_xx - yUyy + yU_x = x ; x>0, y>0

Solution

The standard form of a differential equation is given as

y\'\' + p(t)y\' + q(t)y = f(t)

in this question we have two variables i.e. x and y

x\'\' + yx\' -x = yUyy

the given differential equation is modified in standard form

p(y) = y and q(y) = -1, f(y) = yUyy

Other constraints for standard form are

1) The right hand side must not be having any greater than or equal to sign

2) The right hand side must be positive

which are also satisfied hence the standard form of given PDE is

x\'\' + yx\' -x = yUyy

the given differential equation is modified in standard form

p(y) = y and q(y) = -1, f(y) = yUyy