Substituting u(x, y)=X(x) Y(y) into the partial differential...

Substituting $u(x, y)=X(x) Y(y)$ into the partial differential equation yields $x X^{\prime} Y=y X Y^{\prime} .$ Separating variables and using the separation constant $-\lambda$ we obtain $$\frac{x X^{\prime}}{X}=\frac{y Y^{\prime}}{Y}=-\lambda.$$ When $\lambda \neq 0$ $$x X^{\prime}+\lambda X=0 \quad \text { and } \quad y Y^{\prime}+\lambda Y=0$$ so that $$X=c_{1} x^{-\lambda} \quad \text { and } \quad Y=c_{2} y^{-\lambda}.$$ A particular product solution of the partial differential equation is $$u=X Y=c_{3}(x y)^{-\lambda}.$$ When $\lambda=0$ the differential equations become $X^{\prime}=0$ and $Y^{\prime}=0,$ so in this case $X=c_{4}, Y=c_{5},$ and $u=X Y=c_{6}.$

Substituting $u(x, y)=X(x) Y(y)$ into the partial differential equation yields $x X^{\prime} Y=y X Y^{\prime} .$ Separating variables and using the separation constant $-\lambda$ we obtain
$$\frac{x X^{\prime}}{X}=\frac{y Y^{\prime}}{Y}=-\lambda.$$
When $\lambda \neq 0$
$$x X^{\prime}+\lambda X=0 \quad \text { and } \quad y Y^{\prime}+\lambda Y=0$$
so that
$$X=c_{1} x^{-\lambda} \quad \text { and } \quad Y=c_{2} y^{-\lambda}.$$
A particular product solution of the partial differential equation is
$$u=X Y=c_{3}(x y)^{-\lambda}.$$
When $\lambda=0$ the differential equations become $X^{\prime}=0$ and $Y^{\prime}=0,$ so in this case $X=c_{4}, Y=c_{5},$ and $u=X Y=c_{6}.$

Key Concepts

Separation of Variables

This technique is used to simplify partial differential equations by assuming that the solution can be written as a product of functions, each depending on a single variable. By substituting u(x, y) = X(x)Y(y) into the PDE and dividing appropriately, the PDE is transformed into two ordinary differential equations (ODEs) that depend on x and y separately.

Separation Constant

When the variables are separated, both sides of the resulting equation depend on different independent variables. Since they must be equal for all values of these variables, each side is set equal to a constant, commonly called the separation constant. In this case, it is taken as -?, which facilitates the derivation of two independent ODEs.

First Order Linear Differential Equations

After separation, the resulting equations, such as xX' + ?X = 0 and yY' + ?Y = 0, are first order linear ODEs with variable coefficients. These equations are typically solved using methods like separation of variables or integrating factors.

Special Case Consideration (? = 0 vs ? ? 0)

Solution behavior is examined for different values of the separation constant. For ? ? 0, the ODEs lead to solutions that involve power functions, while for ? = 0, the equations reduce to simpler forms with constant solutions. This distinction is essential in determining the general solution to the original PDE.

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