Math Problem Statement
Solve z=p^2x+q^2x by charpits method
Solution
To solve the partial differential equation using Charpit's method, let's follow these steps:
-
Rewrite the PDE in terms of the variables and :
- The given PDE is:
- In Charpit's method, we have and .
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Compute the Charpit Equations:
- Charpit's equations for a first-order PDE are:
- For our PDE, define .
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Calculate Partial Derivatives of :
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Substitute into Charpit’s Equations:
- The system becomes:
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Solve the System of Equations:
- From , we find that is constant.
- From ,
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Math Problem Analysis
Mathematical Concepts
Partial Differential Equations
Charpit's Method
First-Order PDEs
Formulas
Charpit's equations: dx/F_p = dy/F_q = dz/(p F_p + q F_q) = dp/(-F_x - p F_z) = dq/(-F_y - q F_z)
Theorems
Charpit's Method for Solving First-Order Partial Differential Equations
Suitable Grade Level
Undergraduate (Math or Engineering)
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