Equations of form, Pp+Qq=R are known as Lagrange?s linear equations, named after Franco-Italian mamatician, Joseph-Louis Lagrange (1736-1813).

A solution of a differential equation is any function which satisfies equation, i.e., reduces it to an identity. A solution is also known as integral or primitive.

A differential equation is an equation involving an unknown function y of one or more independent variables x, t, ?? and its derivatives. These are divided into two types, ordinary or partial differential equations.
An ordinary differential equation is a differential equation in which a dependent variable (say ?y?) is a function of only one independent variable (say ?x?).
A partial differential equation is one in which a dependent variable depends on one or more independent variables.

Equations of form Pp + Qq = R , where P, Q and R are functions of x, y, z, are known as Lagrange?s linear equation.

The order of an equation is defined as highest derivative present in equation. Hence, in given equation, \(\frac{?^2 z}{?x^2}-(\frac{?z}{?y})^5+\frac{?^2 z}{?x?y}=0\), order is 2.

The general solution of a differential equation is also called primitive. The solution of a partial differential equation obtained by eliminating arbitrary constants is called a general solution.

If we know n linearly independent solutions u₁(t), …., u_n(t) of a nth order linear homogeneous DE, n general solution of this DE has form:
\(u(t)=c_1 u_1 (t)+?+c_n u_n (t)=?_{k=1}^nc_k u_k (t)\)

Here we use partial differentiation.
\(\frac{\partial \ta}{\partial t}=nt^{n-1}×e^\frac{-r^2}{2t}+t^n×e^\frac{-r^2}{2t}×\frac{r^2}{2t^2}\)
\(\frac{\partial \ta}{\partial t}=nt^{n-1}×e^\frac{-r^2}{2t}+t^n×e^\frac{-r^2}{2t}×\frac{r^2}{2t^2}\)
Substituting from question
\(\frac{\partial \ta}{\partial t}=\frac{n\ta}{t}+\frac{r^2 \ta}{2t^2}\)
\(\frac{\partial \ta}{\partial t}=(\frac{n}{t}+\frac{r^2}{2t^2})\ta \)
\(\frac{\partial \ta}{\partial r}=t^n×e^\frac{-r^2}{2t}×\frac{-r}{t}\)
Substituting from question
\(\frac{\partial \ta}{\partial r}=\frac{-r\ta}{t}\)
\(r^2 \frac{\partial \ta}{\partial r}=\frac{-r^3 \ta}{t}\)
Now substituting values of \(\frac{\partial \ta}{\partial r}\) and \(\frac{\partial \ta}{\partial t}\) in original equation,
\(\left ( \frac{n}{t}+\frac{r^2}{2t^2} \right )\ta = \frac{1}{r^2} \frac{\partial}{\partial r}(\frac{-r^3 \ta}{t})\)
\(\frac{n}{t}=\frac{-1}{t}\)
n=-1.

A differential equation is an equation involving an unknown function y of one or more independent variables x, t, ?? and its derivatives. These are divided into two types, ordinary or partial differential equations.
An ordinary differential equation is a differential equation in which a dependent variable (say ?y?) is a function of only one independent variable (say ?x?).
A partial differential equation is one in which a dependent variable depends on one or more independent variables.

Lagrange?s linear equation contains only first-order partial derivatives which appear only with first power; hence equation is of first-order and first-degree.