Undetermined coefficients, variable separable, and variation of parameters are three techniques used in solving ordinary differential equations.
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- Undetermined coefficients: This technique is used to find a particular solution of a non-homogeneous linear differential equation. The method involves assuming a particular form for the solution, typically a polynomial or an exponential function, and then solving for the unknown coefficients by plugging the assumed solution into the differential equation and solving for the coefficients.
- Variable separable: This technique is used to solve first-order ordinary differential equations that can be written in the form of dy/dx = f(x)g(y), where f(x) and g(y) are functions of x and y, respectively. The method involves separating the variables by dividing both sides of the equation by g(y) and then integrating both sides with respect to x and y separately.
- Variation of parameters: This technique is used to find the general solution of a non-homogeneous linear differential equation. The method involves first finding the general solution of the corresponding homogeneous equation, and then using the method of undetermined coefficients to find a particular solution. The particular solution is then combined with the homogeneous solution to obtain the general solution. The variation of parameters method involves assuming a particular solution in the form of a linear combination of the homogeneous solutions, and then solving for the unknown coefficients by plugging the assumed solution into the differential equation and solving for the coefficients.
