Solving First Order Differential Equations: The y2dy/dx x Case
First-order differential equations are a fundamental topic in mathematics, particularly in topics like physics and engineering. A first-order differential equation involves a function and its first derivative. In this article, we'll explore how to solve the differential equation y2dy/dx x and understand the significance of initial conditions in finding a unique solution. We'll cover the step-by-step reasoning and provide a detailed solution.
Understanding the Differential Equation
Consider the differential equation:
y2 dy/dx x
Here, y is a function of x, and dy/dx is the derivative of y with respect to x. To solve this, we'll follow a systematic approach involving integration and applying the initial condition.
Step-by-Step Solution
Let's start by rewriting the equation:
y2 dy/dx x
Integrate both sides with respect to x to find the general solution:
y2 dy x dx
Now, integrate both sides:
∫ y2 dy ∫ x dx
The integral on the left side is:
(1/3)y3
The integral on the right side is:
(1/2)x2
Adding the constant of integration:
(1/3)y3 (1/2)x2 C
Multiplying through by 6 to clear the fractions:
2y3 3x2 6C
Rearrange to isolate y:
y3 (3/2)x2 3C
Taking the cube root of both sides:
y (3/2x2 3C)1/3
Now, apply the initial condition y0 1. The initial condition means evaluating the function at x 0 and y 1.
Applying the Initial Condition
Substitute x 0 and y 1 into the general solution:
1 (3/2(02) 3C)1/3
This simplifies to:
1 (3C)1/3
Cubing both sides to solve for C:
1 3C
C 1/3
Substituting this value back into the general solution:
y (3/2x2 3(1/3))1/3
Simplifying inside the parentheses:
y (3/2x2 1)1/3
This is the final, unique solution to the differential equation given the initial condition y0 1.
Conclusion
In this article, we discussed the solving method of the first-order differential equation y2dy/dx x and demonstrated the significance of applying initial conditions to find a specific solution. By integrating and solving for the constant C, we were able to obtain the precise formula for y in terms of x.