Methods to Solve Differential Equations
Methods to Solve Differential Equations:

Separation of Variables:
 Applicable to certain firstorder ordinary differential equations (ODEs).
 Involves isolating variables on one side of the equation and integrating both sides separately.
 Typically used for equations that can be written in the form $\frac{dy}{dx}=f(x)g(y)$ or similar separable forms.

Exact Differential Equations:
 Checks if a given firstorder ODE can be written as the total differential of a function.
 If the equation $M(x,y)dx+N(x,y)dy=0$ satisfies the condition $\frac{\mathrm{\partial}M}{\mathrm{\partial}y}=\frac{\mathrm{\partial}N}{\mathrm{\partial}x}$, it's exact.
 Integrating factors are used to make nonexact equations exact.

Integrating Factors:
 Applies to firstorder linear ODEs that are not exact.
 A multiplying factor is introduced to make the equation exact.
 The factor is obtained by solving an integrating factor equation based on the coefficients of the ODE.

Substitution Methods:
 Involves substituting variables or functions to transform the original differential equation into a simpler form.
 Methods like trigonometric substitution, power series substitution, and Bernoulli substitution are used for different types of equations.

Method of Undetermined Coefficients:
 Primarily used for finding particular solutions of nonhomogeneous linear differential equations.
 Assumes a particular form of the solution and determines undetermined coefficients by substituting the form into the differential equation.

Method of Variation of Parameters:
 Applicable to finding particular solutions of nonhomogeneous linear differential equations.
 Involves assuming a solution composed of a general solution of the associated homogeneous equation and using undetermined parameters to find the particular solution.

Laplace Transform:
 Transforms differential equations into algebraic equations using the Laplace transform.
 Solves linear constant coefficient ODEs with initial conditions, simplifying the solving process.

Numerical Methods:
 Includes Euler's method, RungeKutta methods, finite difference methods, and numerical approximation techniques to solve differential equations numerically.
 Used when analytical solutions are challenging or impossible to obtain.

Series Solutions:
 Applies to solving differential equations by representing solutions as power series.
 Particularly useful for equations with singular points or nonconstant coefficients.

Fourier Transform and Other Advanced Techniques:
 Advanced mathematical techniques like Fourier transforms, Green's functions, and special functions (Bessel, Legendre, Hermite) solve specific types of differential equations in physics, engineering, and other disciplines.