Question

Let \(\mathbf{U}=(u, v, w)\) be continuously differentiable with respect to \((x, y)\) and satisfy $$ \begin{aligned} x^{2} y+x y^{2}+u^{2}-(v+w)^{2} &=-3 \\ e^{x+y}-u-v-w &=-2 \\ (x+y)^{2}+u+v+w^{2} &=3 \end{aligned} $$

Short answer

Question: Given that \(\mathbf{U}=(u, v, w)\) is continuously differentiable with respect to \((x, y)\), and using the following system of nonlinear equations: 1. \((v+w)^{2}=x^2 y+xy^2+u^2+3\) 2. \(u+v+w=e^{x+y} - 2\) 3. \(v + u + w^2 = (x+y)^2 - 3\) Find the solution \((x, y, u, v, w)\) that satisfies all the equations. Answer: To solve the given system of nonlinear equations, refer to the given step by step solution to eliminate \(u, v,\) and \(w\), derive a new system of equations with the variables \(x\) and \(y\), solve for \(x\) and \(y\), and finally find the values of \(u, v, w\) based on the obtained values of \(x\) and \(y\).

Step-by-step solution

First equation to expression for \((v+w)^2\)#

From the first equation, we obtain \((v+w)^{2}\) in terms of \(x, y, u\): $$(v+w)^2 = x^2 y+xy^2+u^2+3$$

Second equation to expression for \(u+v+w\)

From the second equation, we obtain \(u+v+w\) in terms of \(x, y\) : $$u+v+w = e^{x+y} - 2$$

Third equation to the expression for \(v + u + w^2\)

Using the third equation, we obtain \(u+v+w^2\) in terms of \(x, y\): $$u+v+w^2 = (x+y)^2 - 3$$

Eliminate \(u\), \(v\), and \(w\) by using the expressions from Steps 1, 2, and 3

Square the expression obtained in Step 2 and compare it with the expression from Step 1: $$(u+v+w)^2 = (e^{x+y} - 2)^2 = x^2 y+xy^2+u^2+3$$ Now, compare the expressions obtained in Steps 2 and 3: $$u+v+w^2 = e^{x+y} - 2 - w = (x+y)^2 - 3$$ Rearrange the above equation: $$w = e^{x+y} - 2 - (x+y)^2 + 3$$ Substitute the expression for \(w\) into the first square equation: $$(u+v+w)^2 = (u+v+(e^{x+y} - 2 - (x+y)^2 + 3))^2 = x^2 y+xy^2+u^2+3$$ which simplifies to: $$(u+v)^2 + 2(u+v)(e^{x+y} - (x+y)^2 + 1) + (e^{x+y} - (x+y)^2 + 1)^2 = x^2 y+xy^2+u^2+3$$ Now, let's compare expressions from Step 1 and Step 3: $$(u+v)^2 = x^2 y+xy^2+u^2+3 - u - v - w^2 = (x+y)^2 - 3 - u - v - w^2$$ We can derive an expression for \((u+v)^2\) in terms of x and y, let this expression be denoted \(F(x, y)\). This will give us a system of equations with only x and y.

Solve the system of equations with only x and y

Our system is: $$\left\{ \begin{aligned} F(x, y) &= x^2 y+xy^2+u^2+3 - u - v - w^2\\ w &= e^{x+y} - 2 - (x+y)^2 + 3 \\ \end{aligned} \right . $$ Solving this system of equations will yield the values of \(x\) and \(y\).

Find values for \(u, v,\) and \(w\)

Once \(x\) and \(y\) have been found, we can substitute their values back into the expressions obtained in Steps 1-3 to retrieve the values for \(u, v\), and \(w\). This gives us the final solution to the problem, \((x, y, u, v, w)\).

Key concepts

Continuously Differentiable Functions

Continuously differentiable functions are cornerstone concepts in real analysis, playing a crucial role in solving systems of equations and finding expressions in terms of variables.

By definition, a function is continuously differentiable if it has derivatives of all orders that are continuous. This means that not only does the function itself smoothly map inputs to outputs without any 'jumps', but its rate of change at every point can be smoothly traced as well. In practical terms, if we represent a function with a curve on a graph, being continuously differentiable ensures that the curve neither has sharp bends nor breaks.

Importance in Systems of Equations

When dealing with systems of equations, continuously differentiable functions facilitate the application of various analytical methodologies, such as the implicit function theorem or various optimization techniques. These properties are especially significant when every variable in the equation system affects every other variable, which requires sensitive and precise calculations for solving the equations.

Usage in the Given Exercise

In our original exercise, we have a system where the function \(\mathbf{U}=(u, v, w)\) is presupposed to be continuously differentiable relative to \(x\) and \(y\). This assumption allows for the step-by-step methods outlined to effectively derive and manipulate the relationships between the variables, ultimately leading to a solution.

System of Equations

A system of equations is a set of multiple equations that share the same variables. The goal is to find a solution that simultaneously satisfies all the equations in the system.

Solving a system of equations can involve several methods, including substitution, elimination, and using matrices. Often, systems of equations emerge in real-world scenarios where multiple conditions must be met at once.

Refined Process for Solving

In the context of our given problem, we start by deriving expressions for combinations of \(u, v, w\) in terms of \(x\) and \(y\). By doing this, the problem systematically progresses towards a simplified form where we have fewer variables to consider. These intermediate expressions are instrumental in forming a new system of equations which ideally contains only \(x\) and \(y\), making it easier to solve.

For instance, the exercise required understanding the relationships presented in the three original equations and manipulating them systematically. This manipulation led us to calculate the expressions in terms of \(x\) and \(y\) only, reducing the complexity of the original system of equations.

Expressions in Terms of Variables

Expression in terms of variables is a phrase referring to the algebraic way of representing the relationships between different variables. In real analysis and algebra, this means rewriting equations to isolate one set of variables as a function of others.

Expressions are the building blocks for forming equations, the clearer the expression, the easier it is to understand and solve the equation. They aid in providing a structured approach to problem-solving by delineating how variables interact with one another.

Strategic Use of Expressions

In solving our exercise, expressions were skillfully formed for \(u+v+w\), \(u+v+w^2\), and \(v+w\), highlighting the interplay of the variables \(u, v, w\) with \(x\) and \(y\). This strategic rewriting was essential to eliminate the variables \(u, v, w\) and reduce the equations to a system involving only \(x\) and \(y\). The isolating of these variables was critical in our step-by-step approach to navigate towards the solution.

Following this methodology, the once seemingly perplexing system of equations transitions into a more straightforward mathematics problem that guides us to the values of \(x\) and \(y\), and sequentially, to the values of \(u, v,\) and \(w\).