lonewolf219 said:Hi, I don't understand why the general solution of 2nd order homogeneous equation is linear? Why is c_1e^(xt)+c_2e^(xt) a linear differential equation? What am I missing here? Any help would be appreciated, I'm struggling a bit understanding the concepts of differential equations...
Understanding the linearity of general solutions of homogeneous equations is important because it allows us to solve for a wide range of variables and constants, making it a versatile tool in many areas of science and mathematics.
A homogeneous solution is one in which all terms have the same degree or power, meaning there are no constant terms. In other words, the solution is not affected by any changes in the independent variable.
A solution to a homogeneous equation is linear if it satisfies the properties of linearity, which include additivity and scalability. Additivity means that when two solutions are added together, the resulting solution is also a valid solution. Scalability means that when a solution is multiplied by a constant, the resulting solution is also a valid solution.
No, a non-linear equation cannot have a linear general solution. A linear general solution implies that the equation is linear, meaning that it follows the properties of linearity. If an equation is non-linear, it does not follow these properties and therefore cannot have a linear general solution.
We can use the linearity of general solutions to solve for unknown constants by plugging in different values for the variables and using the properties of linearity to manipulate the equation and solve for the unknown constants. This method is commonly used in solving systems of linear equations.