Problem 43
Question
First note that $$d(\sqrt{x^{2}+y^{2}})=\frac{x}{\sqrt{x^{2}+y^{2}}} d x+\frac{y}{\sqrt{x^{2}+y^{2}}} d y$$ Then \(x d x+y d y=\sqrt{x^{2}+y^{2}} d x\) becomes $$\frac{x}{\sqrt{x^{2}+y^{2}}} d x+\frac{y}{\sqrt{x^{2}+y^{2}}} d y=d(\sqrt{x^{2}+y^{2}})=d x$$ The left side is the total differential of \(\sqrt{x^{2}+y^{2}}\) and the right side is the total differential of \(x+c\). Thus \(\sqrt{x^{2}+y^{2}}=x+c\) is a solution of the differential equation.
Step-by-Step Solution
Verified Answer
The solution is \(\sqrt{x^2 + y^2} = x + C\).
1Step 1: Understand Given Information
We start with the given identity for the differential of \[d(\sqrt{x^{2}+y^{2}}) = \frac{x}{\sqrt{x^{2}+y^{2}}} \, dx + \frac{y}{\sqrt{x^{2}+y^{2}}} \, dy.\]This represents the total differential of the function \( \sqrt{x^2 + y^2} \).
2Step 2: Reshape Equation
The equation \( x \, dx + y \, dy = \sqrt{x^2 + y^2} \, dx \) is given. We want to express it in a form similar to a differential. Divide through by \( \sqrt{x^2 + y^2} \):\[ \frac{x}{\sqrt{x^2 + y^2}} \, dx + \frac{y}{\sqrt{x^2 + y^2}} \, dy = dx. \]
3Step 3: Compare with Known Differential
The left side of our reshaped equation is now identical to the given total differential \( d(\sqrt{x^2 + y^2}) \). This implies\[d(\sqrt{x^2 + y^2}) = dx.\]
4Step 4: Integrate Both Sides
Integrate both sides to express it in terms of original functions:\[ \sqrt{x^2 + y^2} = x + C, \]where \( C \) is the constant of integration.
5Step 5: State the Solution
The problem implies that \( \sqrt{x^2 + y^2} = x + C \) is a solution to the differential equation. Hence, this is the relationship between the variables that satisfies our initial equation.
Key Concepts
Total DifferentialIntegrationEquation Solving
Total Differential
The concept of a total differential is crucial in multivariable calculus. It provides a way to approximate the change in a function with respect to all its variables. For a function like \ \(f(x, y) = \sqrt{x^2 + y^2}\ \), the total differential \ \(df\ \) represents the change in \ \(f\ \) due to small changes in \ \(x\ \) and \ \(y\ \).
In our exercise, \ \(d(\sqrt{x^2+y^2}) = \frac{x}{\sqrt{x^2+y^2}}dx + \frac{y}{\sqrt{x^2+y^2}}dy\ \) is the total differential of \ \(\sqrt{x^2+y^2}\ \).
This expression essentially tells how the function \ \(\sqrt{x^2+y^2}\ \) changes when you make small changes to \ \(x\ \) and \ \(y\ \).
The differential terms \ \(dx\ \) and \ \(dy\ \) represent infinitesimally small increments in \ \(x\ \) and \ \(y\ \), and their coefficients give the rate of change of \ \(\sqrt{x^2+y^2}\ \) with respect to these variables.
This type of differential analysis facilitates understanding how multivariable functions can change and helps in solving problems that involve small changes across different dimensions.
In our exercise, \ \(d(\sqrt{x^2+y^2}) = \frac{x}{\sqrt{x^2+y^2}}dx + \frac{y}{\sqrt{x^2+y^2}}dy\ \) is the total differential of \ \(\sqrt{x^2+y^2}\ \).
This expression essentially tells how the function \ \(\sqrt{x^2+y^2}\ \) changes when you make small changes to \ \(x\ \) and \ \(y\ \).
The differential terms \ \(dx\ \) and \ \(dy\ \) represent infinitesimally small increments in \ \(x\ \) and \ \(y\ \), and their coefficients give the rate of change of \ \(\sqrt{x^2+y^2}\ \) with respect to these variables.
This type of differential analysis facilitates understanding how multivariable functions can change and helps in solving problems that involve small changes across different dimensions.
Integration
Integration is a key process in calculus used to find solutions to differential equations or to determine areas under curves. In the context of our exercise, integration appears when determining the solution to a differential equation.
Once we have written the differential equation \ \(d(\sqrt{x^2+y^2}) = dx\ \), the next step is to integrate both sides to find the function from its differential.
This process helps us to revert the effects of taking the differential, effectively calculating the original function, possibly up to a constant.
When we perform the integration on both sides:
Once we have written the differential equation \ \(d(\sqrt{x^2+y^2}) = dx\ \), the next step is to integrate both sides to find the function from its differential.
This process helps us to revert the effects of taking the differential, effectively calculating the original function, possibly up to a constant.
When we perform the integration on both sides:
- On the left side, integrating \ \(d(\sqrt{x^2+y^2})\ \) gives us back \ \(\sqrt{x^2+y^2}\ \).
- On the right side, integrating \ \(dx\ \) yields \ \(x + C\ \), where \ \(C\ \) represents the constant of integration.
This constant is crucial because it accounts for the indefinite nature of integration. Every family of curves that differ by a constant is a possible solution to the differential equation.
Equation Solving
Solving a differential equation involves finding a function that satisfies the equation. In this exercise, the challenge is to find a function that fits the given relationships between differentials.
The process begins by recognizing equivalent forms using differentials. We observed that \ \( \frac{x}{\sqrt{x^2+y^2}}dx + \frac{y}{\sqrt{x^2+y^2}}dy = dx\ \) matches the known total differential form.
This insight allows us to equate \ \(d(\sqrt{x^2+y^2})\ =\ dx\ \), leading directly to integration.
After integrating, the solution \ \(\sqrt{x^2+y^2}\ =\ x\ +\ C\ \) offers a relationship that governs \ \(x\ \) and \ \(y\ \) based on their geometric interpretation as a distance. Each \ \(C\ \) determines a specific curve, representing distinct solutions that depend on initial conditions or specific scenarios.
Equation solving in this context showcases how identifiable patterns allow us to derive solutions that satisfy complex differential relations. It's all about transforming, recognizing known forms, and employing calculus methods to unravel solutions that elaborate the behavior of involved variables.
The process begins by recognizing equivalent forms using differentials. We observed that \ \( \frac{x}{\sqrt{x^2+y^2}}dx + \frac{y}{\sqrt{x^2+y^2}}dy = dx\ \) matches the known total differential form.
This insight allows us to equate \ \(d(\sqrt{x^2+y^2})\ =\ dx\ \), leading directly to integration.
After integrating, the solution \ \(\sqrt{x^2+y^2}\ =\ x\ +\ C\ \) offers a relationship that governs \ \(x\ \) and \ \(y\ \) based on their geometric interpretation as a distance. Each \ \(C\ \) determines a specific curve, representing distinct solutions that depend on initial conditions or specific scenarios.
Equation solving in this context showcases how identifiable patterns allow us to derive solutions that satisfy complex differential relations. It's all about transforming, recognizing known forms, and employing calculus methods to unravel solutions that elaborate the behavior of involved variables.
Other exercises in this chapter
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