2nd ODE, Reduction of Order, Basis known

[Darkmisc]

I have a homework problem where I am to find y_2 for a 2nd ODE, with y_1=x.

I'm familiar with the process of:

let y_2 = ux

y_2- = u'x u

y_2'' = 2u' + u''x

substituting these terms into the 2ODE, then letting u' = v.

When integrating v and u' to solve for u, do I need to include integration constants at both steps?

I have a textbook that suggests that integration constants can be made redundant by choosing the 1 and 0 as their values.

However, I have lecture notes which seem to include integration constants in the final solution.

What's the correct approach?

Thanks

 

[HallsofIvy]

It depends upon what you want. If you include the constants of integration you will get something like: u(x)= Cf(x)+ D with C and D constants. Then, from y= ux, your solution is y(x)= (Cf(x)+ D)x= Cf(x)x+ Dx, the general solution to the original equation.

If the equation is linear and you do not include the constants of integration, you will get only u(x)= f(x) so that y(x)= f(x)x. But then you can say "I now have two independent solutions, f(x)x and x, to this linear equation so the general solution is Cf(x)x+ Dx" for any constants C and D. That gives exactly the same solution.

 

[Darkmisc]

Thanks.