Determine the order of the given partial differential equation; also

jernplate8 2021-10-04 Answered
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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Expert Answer

casincal
Answered 2021-10-05 Author has 10059 answers
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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