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Linear Partial Differential Equations: Solving the Equation (d^2-dd'-6d'^2)z=xy
In this article, we will discuss how to solve a specific type of linear partial differential equation (PDE), which is commonly found in mathematics and physics. The equation in question is:
(d^2 - dd' - 6d'^2)z = xy
where z is a function of two independent variables x and y, and d/dx and d/dy are the partial derivatives with respect to x and y, respectively.
What is a Linear Partial Differential Equation?
A linear partial differential equation is a partial differential equation in which the derivative of the dependent variable is proportional to the dependent variable itself. In other words, a linear PDE is an equation of the form:
a(x,y)∂z/∂x + b(x,y)∂z/∂y + cz = f(x,y)
where a, b, and c are functions of x and y, and f is a given function of x and y.
Method of Solution: Separation of Variables
One of the most commonly used methods for solving linear PDEs is the separation of variables method. This method involves assuming that the solution z can be written as a product of two functions, one depending only on x and the other depending only on y.
Let's assume that the solution z can be written as:
z(x,y) = X(x)Y(y)
Substituting this expression into the original equation, we get:
(d^2X - dXd' - 6d'^2X)Y = xy
Since X and Y are functions of x and y, respectively, we can equate the coefficients of x and y to obtain two ordinary differential equations (ODEs):
(d^2X - dXd' - 6d'^2X) = 0
and
Y = xy
Solving these ODEs, we get:
X(x) = Ae^(2x) + Be^(-x) + Ce^(-3x)
and
Y(y) = y
where A, B, and C are arbitrary constants.
General Solution
The general solution to the original equation is, therefore:
z(x,y) = (Ae^(2x) + Be^(-x) + Ce^(-3x))y
This solution satisfies the original equation and can be verified by direct substitution.
Conclusion
In this article, we have demonstrated how to solve a linear partial differential equation using the separation of variables method. The equation (d^2 - dd' - 6d'^2)z = xy was solved, and the general solution was obtained. This method is a powerful tool for solving a wide range of linear PDEs, and its application is essential in various fields of mathematics and physics.
