Determine the order of the given partial differential equation; also

Determine the order of the given partial differential equation; also state whether the equation is linear or nonlinear.
$$\displaystyle{5}^{{{2}}}{u}_{{{x}}}+{u}_{{{t}}}={1}+{u}_{{\times}}$$

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Step 1
The given partial differential equation is $$\displaystyle{5}^{{{2}}}{u}_{{{x}}}+{u}_{{{t}}}={1}+{u}_{{\times}}$$.
Step 2
The order of the partial differential equation is the order of the highest derivative in that equation.
In the given partial differential equation, the highest derivative is $$\displaystyle{u}_{{\times}}$$.
The order of highest derivative $$\displaystyle{u}_{{\times}}$$ is 2.
So, the order of the partial differential equation $$\displaystyle{5}^{{{2}}}{u}_{{{x}}}+{u}_{{{t}}}={1}+{u}_{{\times}}$$.
Step 3
Now determine whether the equation is linear or non linear as follows.
The given equation is $$\displaystyle{5}^{{{2}}}{u}_{{{x}}}+{u}_{{{t}}}={1}+{u}_{{\times}}$$.
Here, the degrees of the partial derivatives $$\displaystyle{u}_{{{x}}},{u}_{{{t}}}\ {\quad\text{and}\quad}\ {u}_{{\times}}$$ are one.
That is, the partial derivatives in the equation occurs linearly.
A partial differential equation in which the degree of dependent variable and its partial derivatives are at most one is said to be linear partial differential equation.
So, the equation $$\displaystyle{5}^{{{2}}}{u}_{{{x}}}+{u}_{{{t}}}={1}+{u}_{{\times}}$$ is linear.
Hence, the equation $$\displaystyle{5}^{{{2}}}{u}_{{{x}}}+{u}_{{{t}}}={1}+{u}_{{\times}}$$ is a second order linear partial differential equation.
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