# Methods to Solve Differential Equations

### Methods to Solve Differential Equations:

1. Separation of Variables:

• Applicable to certain first-order 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 $\frac{dy}{dx}=f\left(x\right)g\left(y\right)$ or similar separable forms.
2. Exact Differential Equations:

• Checks if a given first-order ODE can be written as the total differential of a function.
• If the equation $M\left(x,y\right)dx+N\left(x,y\right)dy=0$ 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 non-exact equations exact.
3. Integrating Factors:

• Applies to first-order 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.
4. 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.
5. Method of Undetermined Coefficients:

• Primarily used for finding particular solutions of non-homogeneous linear differential equations.
• Assumes a particular form of the solution and determines undetermined coefficients by substituting the form into the differential equation.
6. Method of Variation of Parameters:

• Applicable to finding particular solutions of non-homogeneous 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.
7. Laplace Transform:

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

• Includes Euler's method, Runge-Kutta methods, finite difference methods, and numerical approximation techniques to solve differential equations numerically.
• Used when analytical solutions are challenging or impossible to obtain.
9. Series Solutions:

• Applies to solving differential equations by representing solutions as power series.
• Particularly useful for equations with singular points or non-constant coefficients.
10. 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.