Problem 10 - Advanced Engineering Mathematics

Question

In Problems \(1-10\), solve the given differential equation by using an appropriate substitution. $$ x \frac{d y}{d x}=y+\sqrt{x^{2}-y^{2}}, x>0 $$

Step-by-Step Solution

Answer

The solution is \(y = x \sin(\ln{x} + C)\).

Step 1: Analyze the Differential Equation

Observe that the differential equation is given by \(x \frac{d y}{d x}=y+\sqrt{x^{2}-y^{2}}\). We see that there is a square root term that suggests a trigonometric substitution might simplify the problem.

Step 2: Choose an Appropriate Substitution

To simplify the square root term, choose the substitution \(y = x \sin(\theta)\). This choice is motivated by the form \(x^2 - y^2\), which resembles the Pythagorean identity \(1 - \sin^2(\theta) = \cos^2(\theta)\).

Step 3: Differentiate the Substitution

Differentiate the substitution \(y = x \sin(\theta)\) with respect to \(x\):\[ \frac{d y}{d x} = \sin(\theta) + x \cos(\theta) \frac{d \theta}{d x}.\]

Step 4: Substitute Into the Equation

Substitute \(y = x \sin(\theta)\) and \(\frac{d y}{d x}\) into the original differential equation. It becomes:\[ x(\sin(\theta) + x \cos(\theta) \frac{d \theta}{d x}) = x \sin(\theta) + \sqrt{x^2 - (x \sin(\theta))^2}.\] Simplify the square root to \(x \cos(\theta)\).

Step 5: Simplify the Equation

Cancel \(x \sin(\theta)\) from both sides, leaving:\[ x^2 \cos(\theta) \frac{d \theta}{d x} = x \cos(\theta).\] This simplifies to:\[ x \cos(\theta) \frac{d \theta}{d x} = \cos(\theta).\]

Step 6: Separate Variables and Integrate

Assume \(\cos(\theta) eq 0\) and divide both sides by \(\cos(\theta)\):\[ x \frac{d \theta}{d x} = 1.\] This can be further simplified to: \[ \frac{d \theta}{d x} = \frac{1}{x}.\] Integrate both sides with respect to \(x\):\[ \theta = \ln{x} + C,\] where \(C\) is an integration constant.

Step 7: Solve for \(y\)

Recall \(y = x \sin{\theta}\), substitute \(\theta = \ln{x} + C\) back:\[ y = x \sin(\ln{x} + C).\] This is the general solution of the differential equation.

Key Concepts

Trigonometric Substitution

Trigonometric substitution is a powerful technique used to simplify expressions involving square roots of sums or differences of squares. In our differential equation, the term \(\sqrt{x^2 - y^2}\) suggests a form similar to Pythagorean identities, making trigonometric substitution an ideal choice. By substituting \(y = x \sin(\theta)\), the square root simplifies, leveraging the identity \(\sin^2(\theta) + \cos^2(\theta) = 1\). This substitution allows complex expressions to transform into more manageable trigonometric forms. In general, some common substitutions include:

• \(x = a \sin(\theta)\), for \(\sqrt{a^2 - x^2}\)
• \(x = a \tan(\theta)\), for \(\sqrt{x^2 + a^2}\)
• \(x = a \sec(\theta)\), for \(\sqrt{x^2 - a^2}\)

These transformations help reduce the complexity of integration and are particularly useful in solving integrals and differential equations with radical expressions.

Pythagorean Identity

The Pythagorean Identity is a fundamental result in trigonometry, expressing that \(\sin^2(\theta) + \cos^2(\theta) = 1\). This identity is crucial when using trigonometric substitution, as it enables simplification of complex expressions. In the problem, after substituting \(y = x \sin(\theta)\), we leverage the identity to replace \(x^2 - y^2\) with \(x^2 \cos^2(\theta)\).

Using the identity, the square root \(\sqrt{x^2 - (x \sin(\theta))^2}\) simplifies to \(x \cos(\theta)\). This simplification is pivotal since it turns a complicated differential equation into a more straightforward equation, primarily through other trigonometric simplifications and operations.

Integration Techniques

Integration techniques are essential tools for finding solutions to differential equations. In the exercise, integrating after separating variables is a critical step. The separation of variables is possible due to the trigonometric substitution and simplification using the Pythagorean Identity. This allows us to transform the equation into manageable integrals.

In our simplified problem, after the substitutions and algebraic manipulations, the resulting expression was \(\frac{d \theta}{d x} = \frac{1}{x}\). The integration of both sides becomes straightforward:

• \(\int \frac{d \theta}{d x} \, dx = \theta = \ln{x} + C\)

Integration constants are added as solutions often involve families of functions. Proficiency in applying integration techniques requires a good grasp of functions, integration by parts, trigonometric integrals, and substitution.

Variable Separation

Variable separation, or separation of variables, is a method used to solve differential equations by isolating variables on different sides of the equation. This technique requires rewriting the equation so that all terms involving one variable appear on one side, and terms involving the other variable appear on the other side. In our problem, the separated form was achieved by dividing both sides by \( \cos(\theta)\):

• \( x \frac{d \theta}{dx} = 1 \)

The equation \( \frac{d \theta}{dx} = \frac{1}{x} \) was the result of separating the variables. This form made it easy to integrate both sides, leading to the solution \( \theta = \ln{x} + C \). This simple yet powerful technique transforms complex differential equations into integrable forms, streamlining the path to a solution.