One obvious error: a= +1 here , not -1. Yes, you can use "variation of parameters". Also, because the right hand side is a power of x, you could use "undertermined coefficients" although it is slightly harder to "guess" the correct form for a particular solution in such an equation as compared to a "constant coefficients" equation. Finally, the change of variable u= ln(x) will convert this equation to a "constant coefficients" equation.engineer_dave said:Homework Statement
Find the general solution of x^2y" + xy' - y = 1/x
Homework Equations
m(m-1) +am + b = 0 to solve an Euler Cauchy equation
The Attempt at a Solution
a=1 b=-1
m(m-1) -m -1 =0
m^2 - 2m -1 = 0
I just want to know whether the first step is right. And once I find out the values of M, do I use variation of parameters to find the particular solution? Thanks
A differential equation is a mathematical equation that describes the relationship between a function and its derivatives. It involves the use of derivatives, which represent the rate of change of a function, to model real-world phenomena.
The purpose of solving a differential equation is to determine the function that satisfies the equation and accurately predicts the behavior of a system. This can be used in various fields such as physics, engineering, and economics to understand and predict complex systems.
There are several types of differential equations, including ordinary differential equations (ODEs), partial differential equations (PDEs), and stochastic differential equations (SDEs). ODEs involve only one independent variable, while PDEs involve multiple independent variables. SDEs, on the other hand, incorporate random elements into the equation.
The techniques used to solve differential equations depend on the type of equation. Some common methods include separation of variables, substitution, and using the integrating factor. For more complex equations, numerical methods such as Euler's method and the Runge-Kutta method may be used.
Differential equations have various applications in the fields of physics, engineering, finance, and biology. They are used to model and predict the behavior of systems such as population growth, heat transfer, and electrical circuits. They are also used in developing mathematical models for financial markets and biological systems.