Problem 20 - Advanced Engineering Mathematics

Question

Verify that the indicated expression is an implicit solution of the given first-order differential equation. Find at least one explicit solution \(y=\phi(x)\) in each case. Use a graphing utility to obtain the graph of an explicit solution. Give an interval \(I\) of definition of each solution \(\phi\). $$ 2 x y d x+\left(x^{2}-y\right) d y=0 ; \quad-2 x^{2} y+y^{2}=1 $$

Step-by-Step Solution

Answer

The expression is an implicit solution. Explicit solutions are \(y = x^2 \pm \sqrt{x^4 + 1}\) on \(-\infty < x < \infty\).

Step 1: Verify Implicit Solution

To verify that the expression \(-2x^2y + y^2 = 1\) is an implicit solution of the differential equation \(2xy \, dx + (x^2 - y) \, dy = 0\), substitute it into the differential equation. First, find the partial derivatives, \(\frac{dy}{dx}\), implicitly from \(-2x^2y + y^2 = 1\). Differentiate both sides with respect to \(x\):\[-4xy + y^2\frac{dy}{dx} = 0\].Rearranging gives: \(\frac{dy}{dx} = \frac{4xy}{y^2}\).Now substitute this and \(y\) from the implicit function into the differential equation:\(2xy + (x^2 - y)\frac{dy}{dx} = 0\)\(= 2xy + (x^2 - y)\left(\frac{4xy}{y^2}\right) = 0\).Since this satisfies the differential equation, the given condition is verified to be an implicit solution.

Step 2: Solve Implicit Equation for y

The implicit equation is \(-2x^2y + y^2 = 1\). Solve for \(y\) explicitly. Rewrite the equation:\[y^2 - 2x^2y - 1 = 0\],which is a quadratic equation in \(y\) with:- \(a = 1\), \(b = -2x^2\), \(c = -1\).Use the quadratic formula: \[y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \Rightarrow y = \frac{2x^2 \pm \sqrt{4x^4 + 4}}{2}\]which simplifies to:\[y = x^2 \pm \sqrt{x^4 + 1}\].Thus, we have two explicit solutions: \(y_1 = x^2 + \sqrt{x^4 + 1}\) and \(y_2 = x^2 - \sqrt{x^4 + 1}\).

Step 3: Interval of Definition

To determine the interval of definition for \(y = x^2 \pm \sqrt{x^4 + 1}\), observe that the square root function \(\sqrt{x^4 + 1}\) is defined for all real numbers since \(x^4 + 1 > 0\) for all \(x\). Therefore, both explicit solutions are defined for all real \(x\), so the interval of definition for both \(\phi(x)\) solutions is \(I = (-\infty, \infty)\).

Step 4: Graphing the Explicit Solutions

Using a graphing utility, input the explicit solutions \(y_1(x) = x^2 + \sqrt{x^4 + 1}\) and \(y_2(x) = x^2 - \sqrt{x^4 + 1}\). Generate the graphs for these expressions over interval \(x \in (-\infty, \infty)\).Observe the graphical behavior of the solutions to ensure they align with the implicit solution and verify their behavior across the domain.

Key Concepts

Implicit solutions

Implicit solutions are those where the function is not isolated on one side of the equation. Instead, both the dependent and independent variables intermingle on one side. Consider the function \(-2x^2y + y^2 = 1\). This equation is implicit because the variable \(y\) is not isolated; it sits alongside \(x\) without a clear expression for \(y\) in terms of \(x\).
To verify an implicit solution, we differentiate the equation with respect to the independent variable, typically \(x\). For example, when given \(-2x^2y + y^2 = 1\), we differentiate both sides:\[-4xy + 2y\frac{dy}{dx} = 0\].Rearranging provides an expression for \(\frac{dy}{dx}\). By substituting this derivative into the original differential equation, we can check if it holds true. If it does, we confirm that the expression is indeed an implicit solution. This verification step is critical for understanding relationships between variables that aren't straightforward.

Explicit solutions

Explicit solutions are often easier to work with as they clearly express the dependent variable, usually labeled \(y\), in terms of the independent variable, often \(x\). From the implicit expression \(-2x^2y + y^2 = 1\), we can derive explicit solutions by rearranging the terms to solve for \(y\) using algebraic techniques:For instance, rewriting our implicit equation as a quadratic in \(y\):\[y^2 - 2x^2y - 1 = 0\].We can apply the quadratic formula \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find potential solutions, simplifying to:\[y = x^2 \pm \sqrt{x^4 + 1}\].This gives us two explicit solutions \(y_1(x) = x^2 + \sqrt{x^4 + 1}\) and \(y_2(x) = x^2 - \sqrt{x^4 + 1}\). These solutions separate \(y\) from \(x\), making it straightforward to study \(y\)'s dependency on \(x\). Explicit solutions are particularly useful for plotting graphs and analyzing the system's behavior.

Graphing utility

Graphing utilities, such as graphing calculators or software (e.g., Desmos, GeoGebra), can graph mathematical functions to provide a visual representation. These tools are invaluable for illustrating explicit solutions. By inputting explicit solutions like \(y_1(x) = x^2 + \sqrt{x^4 + 1}\) and \(y_2(x) = x^2 - \sqrt{x^4 + 1}\), we can generate graphs that showcase their trajectory and relationship.
Graphing utilities reveal insights that algebraic equations alone might not. They help in verifying if the explicit solutions align with the implicit expression and give information about the function's behavior across its domain. Observing symmetrical characteristics, points of intersection, or where the graphs diverge offers a deeper understanding of the solutions. These visual aids simplify complex equations, making relationships clear at a glance.

Interval of definition

The interval of definition describes where a function or solution holds true or is valid. It is crucial to understand the domain over which a solution is defined. For the explicit solutions \(y = x^2 \pm \sqrt{x^4 + 1}\), the function \(\sqrt{x^4 + 1}\) is defined for all real numbers, as \(x^4 + 1\) is always positive.
Thus, these explicit solutions are valid across all real \(x\), meaning the interval of definition is \((-\infty, \infty)\). Identifying this interval ensures there is no division by zero or taking square roots of negative numbers, which lead to undefined behavior. Understanding the interval assists students in knowing where the function applies, helpful in analyzing behavior and planning further studies or computational approximations.