Finding the general solution of a differential equation is when you find an answer that applies to all values of the variable. In other words, when you find an answer for every value of x, then you have found the general solution.
Differential equations are equations that contain derivatives of variables. There are many variations of differential equations, but all require the same process to find the general solution. You must first find the derivative of the equation, then solve the equation for y, and finally substitute y back into x.
This article will explain in detail how to find the general solution of any ordinary differential equation. It will explain how to find the general solution for three different types of differential equations: linear, first-order nonlinear, and second-order nonlinear.
Write the general solution
Once you have found the integral solution, you can write the general solution by adding an arbitrary constant. An arbitrary constant is any number that is not a function of the variables in the equation.
For example, if we say that x is a constant, then x is not a function of y, i.e. it does not depend on y. We can therefore say that x equals 2 or -2, it will not change the value of the integral.
So, in our case, we can add an arbitrary constant C to our integral and still get the same general solution:
{y|Ax+By=C}={y|A(x+2)y=C}
{\displaystyle \int _{a}^{b}\frac{d}{dx}(Ax+By)=\int _{a}^{b}\frac{d}{dx}}\;(\frac{A}{x+2})_{y|C}}
{\displaystyle \int _{a}^{b}\frac{d}{dx}}\;(\frac{“xy”}{x + 2})_{y|C}}={\displaystyle \int _{{\rm {Arcangle }}}_a^b}\frac{d}{dx}}\;(\underbrace {\left[(x+2)_{y|C}\right]} _{{\rm {Arcangle }}}_a^b)\;{\stackrel {d}{=}}{\stackrel {=1}}}
{\displaystyle \int _{{\rm {Arcangle }}}_a^b}\dfrac{d}{dx}}\;[A(x+2)_{y|C}]+1={A[1]+B[1]where A and B are arbitrary constants}.
Therefore, if you find an integral for A(x+2)y with some arbitrary constants A and B and then add C to it, you will get back our general solution.
In other words: The general solution of Ax+(By)=C is A([x+2])y plus some arbitrary constants.
The trick here is to remember to include all possible constants in your general solution so as not to lose any area under the curve.
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