Introduction to Solving a Differential Equation

Today, we’ll explore a detailed approach to solving a particular type of differential equation. The differential equation in question is:

This equation can be simplified by considering a substitution, which facilitates the solution process. Let’s break it down step-by-step.

Step-by-Step Solution

First, let’s consider the substitution: (w yx). This substitution may help us remove the first derivative and simplify the equation. Applying the substitution, the differential equation transforms into:

This can be simplified to:

Next, we introduce a new variable (y uv). This substitution allows us to separate the equation into simpler components. Using this approach, we can derive the following expressions:

(P frac{2}{x})

(Q -c)

(R 0)

The function (v) can be determined as:

(v e^{-frac{1}{2}int P , dx} e^{-frac{1}{2}int frac{2}{x} , dx} e^{lnleft(frac{1}{x}right)} frac{1}{x})

The functions (Q_1) and (R_1) are then calculated as:

(Q_1 Q - frac{1}{2}frac{dP}{dx} - frac{1}{4}P^2 -c frac{1}{2}left(-frac{2}{x^2}right) - frac{1}{4}left(frac{4}{x^2}right) -c - frac{1}{x^2})

(R_1 R cdot e^{frac{1}{2}int P , dx} 0 cdot e^{frac{1}{2}int frac{2}{x} , dx} 0)

This leads to the simplified differential equation:

Given the transformed equation:

(frac{d^2u}{dx^2} - cu 0)

The solution to this equation is given by:

(u_{CF} C_1e^{-sqrt{c}x} C_2e^{sqrt{c}x})

Finally, the overall solution for the original problem can be expressed as:

Conclusion

This detailed process provides a clear methodology to solve a second-order differential equation. By using the right substitution and method, complex equations can be reduced to simpler, solvable forms.

Keywords

differential equation, advanced method, second-order differential equation