Find The Explicit General Solution To The Following Differential Equation.

Finding the Explicit General Solution to a Differential Equation: A Step-by-Step Guide

Solving differential equations is a cornerstone of applied mathematics, physics, and engineering, allowing us to model everything from population growth to electrical circuits. The ultimate goal in many cases is to find the explicit general solution—a formula that expresses the dependent variable directly in terms of the independent variable and one or more arbitrary constants. This article demystifies the process, focusing on a powerful method for a common class of equations. We will tackle a specific example to illustrate the universal principles you can apply to similar problems. Our example is the first-order differential equation: dy/dx = (x + y)/(x - y). This equation is not separable in its given form, but it belongs to a special category that reveals a clear path to the solution through a clever substitution.

Understanding the Type: Homogeneous Differential Equations

Before diving into steps, identifying the equation's type is crucial. A first-order differential equation is homogeneous if it can be written in the form dy/dx = F(y/x). This means the right-hand side is a function where the numerator and denominator are homogeneous polynomials of the same degree. Let's verify this for our equation.

Take dy/dx = (x + y)/(x - y). Divide every term in the numerator and denominator by x (assuming x ≠ 0): dy/dx = (1 + y/x) / (1 - y/x). Now, let v = y/x, which implies y = vx and, by the product rule, dy/dx = v + x dv/dx. Substituting these into the equation transforms it into a separable equation in terms of v and x. This v-substitution is the standard technique for solving homogeneous equations and is the key to unlocking the explicit general solution.

Step-by-Step Solution to dy/dx = (x + y)/(x - y)

Let's walk through the solution process meticulously.

Step 1: Perform the Homogeneous Substitution

Recognize the equation is homogeneous. Introduce the substitution: v = y/x, therefore y = vx. Differentiate y = vx with respect to x: dy/dx = v + x dv/dx.

Step 2: Rewrite the Original Equation

Substitute y = vx and dy/dx = v + x dv/dx into the original equation: v + x dv/dx = (x + vx) / (x - vx). Simplify the right-hand side by factoring x: v + x dv/dx = [x(1 + v)] / [x(1 - v)] = (1 + v)/(1 - v).

Step 3: Separate the Variables

Our goal is to get all terms involving v on one side and all terms involving x on the other. Rearrange the equation: x dv/dx = (1 + v)/(1 - v) - v. Combine the terms on the right over a common denominator: x dv/dx = [ (1 + v) - v(1 - v) ] / (1 - v). Simplify the numerator: (1 + v) - v(1 - v) = 1 + v - v + v² = 1 + v². Thus: x dv/dx = (1 + v²) / (1 - v). Now, separate the variables: (1 - v)/(1 + v²) dv = (1/x) dx.

Step 4: Integrate Both Sides

We now have a separated equation ready for integration. ∫ (1 - v)/(1 + v²) dv = ∫ (1/x) dx. Split the left integral: ∫ [1/(1 + v²) - v/(1 + v²)] dv = ∫ (1/x) dx. Integrate term by term: ∫ 1/(1 + v²) dv = arctan(v). ∫ v/(1 + v²) dv: Use substitution u = 1 + v², du = 2v dv, so (1/2) du = v dv. This becomes (1/2) ∫ du/u = (1/2) ln|u| = (1/2) ln(1 + v²). Since 1+v² is always positive, we can drop the absolute value. The right side integrates to ln|x| + C, where C is the constant of integration. Putting it all together: arctan(v) - (1/2) ln(1 + v²) = ln|x| + C.

Step 5: Back-Substitute for v

Recall v = y/x. Substitute back: arctan(y/x) - (1/2) ln(1 + (y/x)²) = ln|x| + C. Simplify the logarithmic term: 1 + (y/x)² = (x² + y²)/x². Therefore, ln(1 + (y/x)²) = ln((x² + y²)/x²) = ln(x² + y²) - ln(x²) = ln(x² + y²) - 2 ln|x|. Substitute this into the equation: arctan(y/x) - (1/2)[ln(x² + y²) - 2 ln|x|] = ln|x| + C. Distribute the -1/2: arctan(y/x) - (1

/2) ln(x² + y²) + ln|x| = ln|x| + C. The ln|x| terms cancel out, leaving us with: arctan(y/x) - (1/2) ln(x² + y²) = C.

This is the implicit general solution to the differential equation. It elegantly relates x and y through a transcendental equation involving the arctangent and a logarithmic term. The constant C can be determined if an initial condition is provided, allowing for a particular solution to be found.

Conclusion

Solving the differential equation dy/dx = (x + y)/(x - y) demonstrates the power of recognizing homogeneous equations and applying the appropriate substitution technique. By transforming the original equation using v = y/x, we reduced it to a separable form, integrated both sides, and back-substituted to express the solution in terms of the original variables. The final implicit solution, arctan(y/x) - (1/2) ln(x² + y²) = C, encapsulates the relationship between x and y that satisfies the differential equation. This process not only yields the solution but also reinforces the importance of methodical problem-solving in differential equations, a skill that is invaluable in both theoretical and applied mathematics.