giacomh said:Homework Statement
Find the general solution to:
yii +y=sec2(t)
The Attempt at a Solution
I found the particular solution, which is
Yp=-sec(t)cos(t)+ln|sec(t)+tan(t)|sin(t)
Is the general solution just y(t)=C1cos(t)+C2sin(t)+Yp?
I just can't find an example of an inhomogeneous problem with complex numbers to verify this.
A general solution to an inhomogeneous equation is a mathematical expression that satisfies the equation for all possible values of the variables involved. It includes both the particular solution (a specific solution for given initial conditions) and the homogeneous solution (a solution to the associated homogeneous equation).
A particular solution is a specific solution to an inhomogeneous equation that takes into account the given initial conditions. A homogeneous solution is a solution to the associated homogeneous equation, which is obtained by setting all constant terms in the equation to zero.
To find the general solution to an inhomogeneous equation, you need to first solve the associated homogeneous equation by setting all constant terms to zero. Then, you can find a particular solution by using a method such as undetermined coefficients or variation of parameters. Finally, the general solution is the sum of the particular solution and the homogeneous solution.
The most commonly used methods for finding a particular solution to an inhomogeneous equation are the method of undetermined coefficients and the method of variation of parameters. Both methods involve using a combination of known functions (such as polynomials, exponential functions, and trigonometric functions) to find a particular solution.
No, the general solution to an inhomogeneous equation is not always unique. The general solution may include a constant term, which can take on any value. Therefore, there can be an infinite number of general solutions to an inhomogeneous equation.