Differential equations contain derivatives, solving the equation involves integration (to get rid of the derivatives). We will cover the most common methods to solve ODE’s: linear, separable and Bernoulli.
- Linear first order equation is an ODE of the form y'(x)+p(x)y(x)=q(x)
- Separable equation is an ODE of the form N(y)dy=M(x)dx
- Bernoulli equation is an ODE of the form y'+p(x)y=q(x)y^n
You have first to identify the ODE type (can be tricky); then simply follow the steps as described below.
Here’s an example of a separable equation (click here):
Simply solve by integrating both sides of the equation:
Here’s an example of a linear first order equation (click here):
Steps to find the integration factor:
Here’s an example of a Bernoulli equation (click here):
In the next post we will take a closer look at each of the ODE types.
Cheers,
Michal
FAQs
Slopes is an interactive environment for exploring graphical solutions to ordinary differential equations. Slopes consists of five activities with preloaded examples to get you started and the ability to input your own content.
Can Symbolab solve ODEs? ›
It can solve ordinary linear first order differential equations, linear differential equations with constant coefficients, separable differential equations, Bernoulli differential equations, exact differential equations, second order differential equations, hom*ogenous and non hom*ogenous ODEs equations, system of ODEs, ...
What is the general solution of the ODE? ›
Theorem The general solution of the ODE a(x) d2y dx2 + b(x) dy dx + c(x)y = f(x), is y = CF + PI, where CF is the general solution of hom*ogenous form a(x) d2y dx2 + b(x) dy dx + c(x)y = 0, called the complementary function and PI is any solution of the full ODE, called a particular integral.
What is the explicit particular solution? ›
An explicit solution is any solution that is given in the form y=y(t) y = y ( t ) . In other words, the only place that y actually shows up is once on the left side and only raised to the first power. An implicit solution is any solution that isn't in explicit form.
What is the best tool to solve differential equations? ›
Maple is the world leader when it comes to solving differential equations, finding closed-form solutions to problems no other system can handle.
What is the free app that solves math equations? ›
Photomath is known worldwide for helping millions of learners to learn, practice, and understand math – one step at a time. Scan any math problem with the Photomath app to get step-by-step explanations with accurate solutions and a variety of teacher-approved methods.
Can Wolfram Alpha solve systems of differential equations? ›
The Wolfram Language's differential equation solving functions can be applied to many different classes of differential equations, automatically selecting the appropriate algorithms without needing preprocessing by the user.
What happened to Symbolab? ›
In 2020, the company was acquired by American educational technology website Course Hero.
Can Mathematica solve ODE? ›
Mathematica features two functions for solving ODEs: DSolve and NDSolve. DSolve is used when the user wishes to find the general function or functions which solve the differential equation, and NDSolve is used when the user has an initial condition.
What is the difference between PDE and ODE? ›
Ordinary differential equations or (ODE) are equations where the derivatives are taken with respect to only one variable. That is, there is only one independent variable. Partial differential equations or (PDE) are equations that depend on partial derivatives of several variables.
A first order differential equation is an equation of the form F(t,y,˙y)=0.
What does PDE stand for in math? ›
Partial Differential Equation (abbreviated in the following as PDE in both singular and plural usage) is an equation for an unknown function of two or more independent variables that involves partial derivatives.
How to solve a differential equation explicitly? ›
Finding the derivative explicitly is a two-step process: (1) find y in terms of x, and (2) differentiate, which gives us dy/dx in terms of x. Finding the derivative implicitly is also two steps: (1) differentiate, and (2) solve for dy/dx. This method may leave us with dy/dx in terms of both x and y.
What makes a differential equation autonomous? ›
In mathematics, an autonomous system or autonomous differential equation is a system of ordinary differential equations which does not explicitly depend on the independent variable. When the variable is time, they are also called time-invariant systems.
How to tell if an equation is implicit or explicit? ›
An implicit function is one that has several variables, one of which is a function of the other set of variables. An explicit function is one in which the dependent variable can be written explicitly in terms of the independent variable. f(x, y) = 0 is the general form of an implicit function.
What are the applications of solving differential equations? ›
Ordinary Differential Equations are used to calculate the movement or flow of electricity, motion of an object to and fro like a pendulum, to explain thermodynamics concepts. Also, in medical terms, they are used to check the growth of diseases in graphical representation.
What software is used to solve partial differential equations? ›
The FEniCS computing platform
FEniCS is a popular open-source computing platform for solving partial differential equations (PDEs) with the finite element method (FEM). FEniCS enables users to quickly translate scientific models into efficient finite element code.
