When dealing with a system of ordinary differential equations (ODEs), the linearity principle often helps in formulating a solution. In this article, we will focus on solving the ODE y’ - xy -x and explore the methods and principles involved in finding its solution. Understanding these techniques is crucial for those working in fields such as physics, engineering, and applied mathematics. The solution involves a series of algebraic manipulations and integration, which we will walk through step-by-step.
Solution Steps
The given differential equation is:
y’ - xy -x
This can be rearranged to:
dy/dx xy - x
First, we can factor out the x on the right-hand side:
dy/dx x(y - 1)
Next, we separate the variables by dividing both sides by y - 1
dy/y - 1 x dx
Integrating Each Side
By integrating both sides of the equation, we get:
ln(y - 1) ∫ x dx
This integral evaluates to:
ln(y - 1) x^2/2 c'
Exponentiating both sides to solve for y, we obtain:
y - 1 e^[x^2/2 c'] e^[x^2/2] * e^[c']
Let K e^[c'], then:
y - 1 K e^[x^2/2]
Thus, the general solution is:
y 1 K e^[x^2/2]
Verification of the Solution
To verify the solution, we take the derivative of y with respect to x.
y' d/dx (1 K e^[x^2/2])
This yields:
y' K e^[x^2/2] * x
Now, substituting y 1 K e^[x^2/2] into the original equation:
y' - xy -x
We substitute y' K e^[x^2/2] * x and y 1 K e^[x^2/2]:
K e^[x^2/2] * x - (1 K e^[x^2/2])x -x
Simplifying the equation, we get:
K e^[x^2/2] * x - x - K e^[x^2/2] x -x
This simplifies to:
-x -x
Which confirms that the solution is correct.
Conclusion
In summarizing this process, solving the ODE y’ - xy -x involves steps of rearrangement, variable separation, integration, and verification. The key principle here is the linearity of the differential equation, which simplifies the process. Understanding these methods not only solves the given equation but also serves as a foundation for tackling more complex ODEs.