Solving Tough 2nd Order DE: x*y'' + y' - a*y = 0

  • Thread starter hawaiifiver
  • Start date
  • Tags
    2nd order
In summary, a second order differential equation is a mathematical equation that involves a function and its derivatives up to the second order. It is called a "tough" equation because it cannot be solved easily using traditional methods. The general method for solving these equations involves reducing them to a simpler form and then using traditional techniques. These equations are important in science as they are used to model physical phenomena and have real-world applications in various fields.
  • #1
hawaiifiver
56
1
T

Homework Statement



Solve x*y'' + y' - a*y = 0

where a > 0


Homework Equations



Not sure what's relevant here. See Below.

The Attempt at a Solution



I think this can be solved by changing the independent variable. I tried x = √t, x = 1/t, x = ln(t), x = exp(t) but these seem to make the transformed DE more complicated.

I don't think I can use Reduction of Order or Variation of parameters here. I don't have any solution y_1.
 
Physics news on Phys.org
  • #2
I think this is a non-homogeneous equation.I'm not sure how to proceed. Any help would be appreciated!
 

FAQ: Solving Tough 2nd Order DE: x*y'' + y' - a*y = 0

What is a second order differential equation?

A second order differential equation is a mathematical equation that involves a function and its derivatives up to the second order. It can be written in the form of x*y'' + y' + a*y = 0, where x is the independent variable, y is the dependent variable, and a is a constant.

Why is it called a "tough" second order differential equation?

A "tough" second order differential equation is one that cannot be solved easily using traditional methods. In this case, the presence of multiple derivatives and the constant term makes it a challenging equation to solve.

What is the general method for solving a tough second order differential equation?

The general method for solving a tough second order differential equation involves reducing the equation to a simpler form, such as a first order equation, by using substitution or other techniques. Then, the equation can be solved using traditional methods such as separation of variables or integrating factors.

What is the importance of solving second order differential equations in science?

Second order differential equations are used to model a wide range of physical phenomena in science, including motion, heat transfer, and population growth. Being able to solve these equations allows scientists to make predictions and understand the behavior of complex systems.

Are there any real-world applications of solving "tough" second order differential equations?

Yes, there are many real-world applications of solving tough second order differential equations. For example, in physics, these equations are used to describe the motion of objects under the influence of forces. In engineering, they are used to design structures and optimize processes. In biology, they are used to model population growth and disease spread. The ability to solve these equations is crucial for advancing our understanding of the natural world.

Similar threads

Obtain a nonzero solution of ##y''-4y'+x^2(y'-4y)=0## by inspection
Replies
7
Views
1K
Understanding Wronskian of solutions to homoeneous 2nd order linear DE
Replies
2
Views
806
Why is this not a general solution to this nonlinear DE?
Replies
5
Views
788
Solving differential equation (t^2-1)y''-6y=1
Replies
2
Views
578
Proof given ##x < y < z## and a twice differentiable function
Replies
8
Views
965
Prove if ##x<0## and ##y<z## then ##xy>xz## (Rudin)
Replies
2
Views
1K
Variation of parameter VS Undetermined Coefficients
Replies
7
Views
1K
Understanding calculation of 2nd order LTI DE response to step input
Replies
8
Views
868
Prove that solutions to autonomous first order DE can't have local maxima
Replies
2
Views
884
Find f(x,y) given partial derivative and initial condition
Replies
6
Views
1K

Hot Threads

Recent Insights

Back
Top