Take y2 and plug it back into the original differential equation and see if it is satisfied.Mark Brewer said:
No, the problem said to use the method of reduction of order.Mark Brewer said:
You mean x2y" + xy' + (x2 - 1/4)y = 0, don't you? As Vela pointed out, the given y1 does not satisfy the equation you gave. Perhaps it satisfies this one. I haven't checked.Mark Brewer said:
So you start by looking for a solution of the form [itex]y_2= u(x)x^{1/2}cos(x)[/itex]
Mark Brewer said:
A second order ordinary differential equation (ODE) is an equation that contains a second derivative of an unknown function. A homogenous equation is a type of ODE in which all terms are of the same order and can be expressed in terms of the dependent variable and its derivatives.
The general solution to a second order ODE homogenous equation is a function that satisfies the equation for all possible values of the independent variable. It contains two arbitrary constants, which can be determined by applying initial conditions.
To solve a second order ODE homogenous equation, you can use the method of undetermined coefficients or the method of variation of parameters. Both methods involve finding a particular solution and adding it to the general solution to get the complete solution.
Second order ODE homogenous equations are important in science because they can model a wide range of physical phenomena, such as oscillations, vibrations, and growth or decay processes. They are also used in many areas of engineering and physics, including mechanics, electricity and magnetism, and quantum mechanics.
Second order ODE homogenous equations can be applied to real-world problems by using mathematical techniques to solve them and obtain solutions that describe the behavior of a system over time. These solutions can then be used to make predictions and analyze the dynamics of the system, which can be helpful in fields such as biology, economics, and environmental science.