ismail amre said:Homework Statement
dy/dx = (y2-1)/(x2-1)
Homework Equations
The Attempt at a Solution
I tried to solve it by method of separation of variable and i reached
dy/(y2-1)=dx/(x2-1)
then by integrating both side i should have this answer which i got from the solution manual which is ln (y-1) - ln (y+1) = ln (x-1) - ln(x+1) .
i understand all the solution except the integration of any side of this equation dy/(y2-1)=dx/(x2-1)
LCKurtz said:Use partial fractions: 1/(x2-1) = A/(x+1) + B/(x-1)
A differential equation is a mathematical equation that relates the rate of change of a variable to the value of the variable itself. It is commonly used to model physical and natural phenomena in various fields such as physics, engineering, and economics.
There are several types of differential equations, including ordinary differential equations, partial differential equations, and stochastic differential equations. Ordinary differential equations deal with functions of a single variable, while partial differential equations involve functions of multiple variables. Stochastic differential equations incorporate randomness and are used in the study of systems with uncertain behavior.
The method for solving a differential equation depends on its type and complexity. Some equations can be solved analytically, meaning a closed-form solution can be found using mathematical techniques. Other equations may require numerical methods, such as Euler's method or the Runge-Kutta method, to approximate a solution. In some cases, differential equations cannot be solved exactly and must be solved using computer software.
Initial conditions are the values of the dependent variable and its derivatives at a specific starting point. These conditions are used to determine the unique solution to a differential equation. Boundary conditions are constraints on the dependent variable at certain points in the domain. They are often used to determine the specific form of the solution to a differential equation.
Differential equations have many real-world applications, such as modeling population growth, predicting the spread of diseases, analyzing electrical circuits, and predicting the motion of celestial objects. They are also used in fields such as economics, chemistry, and biology to model various phenomena and make predictions.