Solving a Differential Equation: A Comprehensive Guide
Solving a Differential Equation: A Comprehensive Guide
In this guide, we will explore how to solve a specific type of differential equation. Specifically, we will deal with the equation:
Given Differential Equation:
frac;d^2y frac;dx^2 (1 (dyfrasl;dx)^2)
Step-by-Step Solution
Step 1: Variable Substitution
First, let's introduce a substitution to simplify the given differential equation. Consider:
u frac;dyfrasl;dx
Substituting this into the given equation, we get:
frac;dufrasl;dx 1 u^2
Step 2: Separation of Variables
This new equation is now in the form of a separable differential equation. We can separate the variables u and x as follows:
Separable Differential Equation:
frac;dufrasl{(1 u^2)} dx
Step 3: Integration
To integrate the left-hand side, we require a trigonometric substitution for the integrand involving 1 u^2. The appropriate substitution is:
u tanθ, du sec^2θ dθ
Substituting these into the integral, we have:
int; frac;sec^2θfrasl;(1 tan^2θ) dθ int; dx
Since 1 tan^2θ sec^2θ, the integral simplifies to:
int; dθ x c_1
Thus, we get:
θ x c_1
Since u tanθ, we further simplify:
u tan(x c_1)
Step 4: Reversing the Substitution
Recall that u frac;dyfrasl;dx. So, substituting back, we have:
frac;dyfrasl;dx tan(x c_1)
To find y, we integrate both sides:
y int; tan(x c_1) dx
Using the integral of tanθ, we get:
y ln|sec(x c_1)| c_2
Therefore, the general solution to the differential equation is:
y ln|sec(x c_1)| c_2
Conclusion
In this guide, we detailed the process of solving a differential equation by using substitution and separation of variables. The final solution involves the natural logarithm of the secant function and a constant of integration. This example showcases the application of first-order differential equations and separable differential equations, which are fundamental concepts in mathematical physics and engineering.