Finding the Second Solution to a Homogeneous Second Order DE

In summary, the conversation is about finding a solution to the given differential equation and the difficulties in doing so. The method of reduction of order is mentioned and one person suggests guessing and checking as a potential solution strategy. The question of whether these types of equations are common in differential equations and the possibility of using numerical approximation is also brought up.
  • #1
maxsthekat
55
0

Homework Statement


4xy'' + 2y' + y = 0


2. The attempt at a solution

In class, we were given that y1 = c1Cos([tex]\sqrt{}x[/tex]). We then used reduction of order to figure out the other solution

Yet, I've been trying to figure out, is how do you get y1 in the first place? To me, it doesn't seem like a Cauchy-Euler equation, I don't think I can apply annihilators to it (since it's homogeneous), and since the coefficients aren't constant, it doesn't look as if I can apply variation of parameters.

Can anyone point me in the right direction?

Thanks!

-Max
 
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  • #2
If you had the method to find y1 systematically, they wouldn't give it to you. Your best bet sometimes is just guess and check (best bet doesn't mean good bet!)
 
  • #3
How would you even begin to know to try that with the square root of x as an argument of cosine? ...Do these sorts of equations often pop up in DE? If so, is numerical approximation the rule of the land?
 

FAQ: Finding the Second Solution to a Homogeneous Second Order DE

What is a second order homogeneous differential equation?

A second order homogeneous differential equation is a mathematical equation that involves a function, its first derivative, and its second derivative. The equation is "homogeneous" because all the terms involve the function and its derivatives, and there are no other independent variables.

What is the general form of a second order homogeneous differential equation?

The general form of a second order homogeneous differential equation is y'' + p(x)y' + q(x)y = 0, where p(x) and q(x) are functions of x.

How do you solve a second order homogeneous differential equation?

To solve a second order homogeneous differential equation, you can use the method of undetermined coefficients or the method of variation of parameters. Both methods involve finding a particular solution and a complementary solution, which are then combined to form the general solution.

What is the difference between a first order and a second order homogeneous differential equation?

The main difference between a first order and a second order homogeneous differential equation is the number of derivatives involved. A first order equation involves the first derivative of a function, while a second order equation involves both the first and second derivatives of a function.

Why are second order homogeneous differential equations important in physics and engineering?

Second order homogeneous differential equations are important in physics and engineering because they can be used to model physical systems that involve acceleration and velocity. For example, the motion of a pendulum or a spring-mass system can be described by a second order homogeneous differential equation.

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