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how to get differential equation from general solution ?

**Finding the Differential Equation from a General Solution**

In this article, we will explore the process of finding the differential equation from a given general solution. This is a fundamental concept in differential equations, and it is essential to understand how to approach this problem.

**What is a General Solution?**

Before we dive into the process of finding the differential equation, let’s first define what a general solution is. A general solution is a solution to a differential equation that contains one or more arbitrary constants. These constants represent the initial conditions or boundary conditions of the problem.

**Example Solutions**

Let’s consider a few example solutions that require their differential equations to be found:

* y = ax^2 + bx + c
* y^2 = 4ax
* x^2 – 2xy + y^2 = a^2

**Finding the Differential Equation**

To find the differential equation from a general solution, we need to differentiate the general solution with respect to x until we get back to the original function. This process is called “differentiation”.

* Let’s consider the first example, y = ax^2 + bx + c. Differentiating this function with respect to x, we get:
+ dy/dx = 2ax + b
+ d^2y/dx^2 = 2a
* For the second example, y^2 = 4ax, differentiate both sides with respect to x:
+ 2y(dy/dx) = 4a
+ dy/dx = 2a/y
* For the third example, x^2 – 2xy + y^2 = a^2, differentiate both sides with respect to x:
+ 2x – 2y(dy/dx) + 2y(dy/dx) = 0
+ dy/dx = x/y

**Key Point**

The order of the differential equation should match the number of arbitrary constants in its general solution. One parameter family solution gives rise to a first-order differential equation, while a two parameter family solution gives rise to a second-order differential equation.

**Wolfram Alpha Widget**

Wolfram Alpha provides a widget that can help you find the differential equation from a general solution. This widget can be used to input the general solution and automatically generate the corresponding differential equation.

**Conclusion**

In this article, we have learned how to find the differential equation from a general solution. This process involves differentiating the general solution with respect to x until we get back to the original function. The order of the differential equation should match the number of arbitrary constants in its general solution. With the help of Wolfram Alpha, this process can be simplified and automated.

**References**

* https://math.stackexchange.com/questions/1415905/finding-the-differential-equation-given-a-solution
* https://www.youtube.com/watch?v=Gd9QXz9husg
* https://www.statisticshowto.com/differential-equations/general-solution-of-differential-equation/
* https://www.studysmarter.us/explanations/math/calculus/general-solution-of-differential-equation
* https://glennolasco.com/finding-the-differential-equation-from-a-general-solution/

I hope this article has been helpful in understanding how to find the differential equation from a general solution. Happy learning!