Problem 1 - Advanced Engineering Mathematics

Question

Letting \(y=u x\) we have $$\begin{aligned} (x-u x) d x+x(u d x+x d u) &=0 \\ d x+x d u &=0 \\ \frac{d x}{x}+d u &=0 \\ \ln |x|+u &=c \\ x \ln |x|+y &=c x. \end{aligned}$$

Step-by-Step Solution

Answer

The solution is \(x \ln |x| + y = cx.\)

Step 1: Rearrange the Differential Equation

We begin by simplifying the given expression \[(x-u x) d x+x(u d x+x d u) =0.\] Notice that \(y = u x\) implies \(x - u x = x(1-u)\). Therefore, the equation becomes \[x(1-u) d x + x(u d x + x d u) = 0.\] Simplifying gives \[x d x - u x d x + x u d x + x^2 d u = 0,\] which combines to \[x d x + x^2 d u = 0.\] Divide through by \(x\) factoring common terms which results in \[d x + x d u = 0.\]

Step 2: Simplify and Separate the Variables

From the expression \[d x + x d u = 0,\] divide by \(x\) to separate the variables:\[\frac{d x}{x} + d u = 0.\]This simplifies the expression, allowing you to integrate with respect to each variable independently.

Step 3: Integrate Both Sides

Integrate each side of the separated equation \[\frac{d x}{x} + d u = 0\] with respect to its variable:\[\int \frac{d x}{x} + \int d u = \int 0.\]The integrals evaluate to:\[\ln |x| + u = c,\] where \(c\) is the constant of integration.

Step 4: Solve for y in terms of x

Since we've defined \(y = ux\), replace \(u\) in terms of \(x\) from the integration result:\[u = c - \ln|x|.\]Substitute back to find \(y\):\[y = (c - \ln|x|)x.\]Rearrange this to express the original result:\[x\ln|x| + y = cx.\]

Key Concepts

Integration Techniques

Differential equations often require integration techniques to find solutions. In this problem, separating variables allows us to set up integrals for each part of the equation. By integrating, we find relationships between variables that help solve the problem.

In particular, the equation given transforms into \[\frac{d x}{x} + d u = 0, \]which separates into two simpler integrals:

• \(\int \frac{d x}{x} = \ln |x|\)
• \(\int d u = u\)

After integrating, we obtain \[\ln |x| + u = c,\]where \(c\) is the constant of integration that arises due to indefinite integration. This integral results in a natural logarithm because of the \(\frac{d x}{x}\) term.

Overall, integration techniques transform difficult differential equations into solvable expressions.

Separation of Variables

Separation of variables is a simple, yet effective technique for solving differential equations. It relies on separating the components in such a way that each side of the equation involves just one variable. Here's how it works in this problem:

We begin with \[d x + x d u = 0.\]By dividing everything by \(x\), we separate the variables: \(\frac{d x}{x} + d u = 0.\)This separation enables each variable to have its own side of the equation. This makes the equation easier to handle because it translates into separate integrals for \(x\) and \(u\), which leads to finding their relationships through integration.

Separation of variables requires careful algebraic manipulation to ensure clear splitting, which turns complex interactions between variables into manageable parts before integration.

Constant of Integration

The constant of integration is a fundamental part of any indefinite integral. After integration, the constant \(c\) represents an infinite number of possible solutions. Each solution corresponds to a different initial condition or specific context of the problem.

In our integration \[\int \frac{d x}{x} + \int d u = \int 0,\]we find \[\ln |x| + u = c.\]This \(c\) is crucial for capturing all potential solutions, since differential equations have multiple solutions unless a specific condition is applied.

Understanding the role of \(c\) helps in properly interpreting solutions of differential equations, and reflects the diversity and flexibility in real-world problems.