A constrained differential probelm

In summary, a constrained differential problem is a mathematical problem that involves finding the best solution to a function while also satisfying a set of constraints. Common examples include optimization problems in various fields, and they are typically solved using mathematical techniques such as Lagrange multipliers. The main challenges in solving these problems are their complexity and ensuring a global optimum solution. Constrained differential problems are also used in scientific research for modeling and understanding complex systems.
  • #1
Leo Liu
353
156
Homework Statement
.
Relevant Equations
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Define that $$w(x,y,z)=zxe^y+xe^z+ye^z$$
1614266061848.png

So the constraint equation is ##x^2y+y^2x=1##. And its differential is ##dy=-\frac{2xy+y^2}{2xy+x^2}##.

However, the solution plugs in ##z=0## when computing ##\frac{\partial w}{\partial x}## as shown in the screenshot below. While I understand that ##dz=0##, I can't see why ##z=0##. Could anyone explain?
1614266672731.png
 
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  • #2
Can you show us the first parts of the problem?
 
  • #3
Office_Shredder said:
Can you show us the first parts of the problem?
Sure, Here it is.
1614301330031.png
 

FAQ: A constrained differential probelm

What is a constrained differential problem?

A constrained differential problem is a mathematical problem that involves finding a solution to a system of differential equations subject to certain constraints. These constraints can be in the form of boundary conditions, initial conditions, or other restrictions on the variables in the system.

How is a constrained differential problem different from a regular differential problem?

A regular differential problem involves finding a solution to a system of differential equations without any constraints. In a constrained differential problem, the solution must satisfy certain conditions or constraints in addition to the differential equations.

What are some common methods for solving constrained differential problems?

Some common methods for solving constrained differential problems include the Lagrange multiplier method, the penalty method, and the augmented Lagrangian method. These methods involve incorporating the constraints into the objective function and solving the resulting optimization problem.

What are some real-world applications of constrained differential problems?

Constrained differential problems have many applications in science and engineering, such as in the design of control systems, optimization of chemical processes, and modeling of physical systems. They are also used in economics, finance, and other fields to solve complex problems with constraints.

What are the challenges in solving constrained differential problems?

One of the main challenges in solving constrained differential problems is finding an efficient and accurate method that can handle both the differential equations and the constraints. The choice of method can also depend on the specific problem and the type of constraints involved. Additionally, the complexity of the problem can increase with the number of constraints, making the solution process more challenging.

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