Second order DE, reduction of order method?

In summary, the task is to find the general solution for the second order differential equation x2y''-xy'-3y=0 using the given solutions y1=1/x and y2=x3. The approach suggested is to use the method of reduction of order, but the only theory available in the text is for constant coefficients. However, it is possible to use the superposition of homogeneous equations to obtain a general solution of y=c1(1/x)+c2(x^3). This solution may seem too simple, but it is correct.
  • #1
ProPatto16
326
0

Homework Statement



find general solution of x2y''-xy'-3y=0

Homework Equations



two solutions are given, y1=1/x and y2=x3

The Attempt at a Solution



i think this is a reduction of order question? the only theory i can find in my text for second order DE relates to constant coefficients, and obvioulsy these ones arent. anyone have some method for reduction of order? or am i going about this the wrong way?

thanks
 
Physics news on Phys.org
  • #2
hold up... by superposition of homogeneous equations...

with y1 and y2 solutions, then a general solution would just be y=c1(1/x)+c2(x^3) ??

but that seems too easy...
 

1. What is the reduction of order method for second order differential equations?

The reduction of order method is a technique used to solve second order differential equations when one solution is already known. It involves substituting a new variable for the known solution, reducing the differential equation to a first order equation, and then solving for the new variable.

2. When is the reduction of order method useful?

The reduction of order method is useful when we know one solution to a second order differential equation but need to find another independent solution. This often happens when solving boundary value problems or when one solution is easy to find, such as a polynomial or exponential function.

3. How does the reduction of order method work?

The reduction of order method works by substituting a new variable u for the known solution y1 and rewriting the original differential equation in terms of u. This reduces the equation to a first order linear differential equation, which can be solved using standard techniques. Finally, the solution for u is substituted back into the original equation to find the second solution y2.

4. Are there any restrictions on using the reduction of order method?

Yes, there are some restrictions on using the reduction of order method. It can only be used when the second order differential equation is linear and homogeneous, meaning it can be written in the form y'' + p(x)y' + q(x)y = 0. Additionally, the known solution y1 must be independent of the other solution y2.

5. Can the reduction of order method be applied to higher order differential equations?

Yes, the reduction of order method can be extended to higher order differential equations. However, it becomes more complicated as the order of the equation increases. It is also important to note that more than one known solution may be required for higher order equations, making it less practical compared to other methods such as variation of parameters or the Wronskian method.

Similar threads

  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
5
Views
191
  • Calculus and Beyond Homework Help
Replies
2
Views
2K
  • Calculus and Beyond Homework Help
Replies
3
Views
2K
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
9
Views
980
  • Calculus and Beyond Homework Help
Replies
1
Views
2K
  • Calculus and Beyond Homework Help
Replies
8
Views
116
Back
Top