- #1
mathwizarddud
- 25
- 0
[tex]y''+xy'+y=0[/tex]
PS:
Is there a way to prove that one has found the "most" general solution by only using a particular method (e.g., integrating factors), with the possiblity of obtaining a different general solution using a different method?
For example, using integrating factor in the ODE
y' + x²y = x
exp(x³/3) y' + exp(x³/3) x²y = exp(x³/3) x
d(exp(x³/3) y) = exp(x³/3)x dx
Integrate both sides:
[tex]exp(x^3/3) y = \int \exp(x^3/3)x\ dx[/tex] or
[tex]y = \frac{\int \exp(x^3/3)x\ dx}{\exp(x^3/3)}[/tex]
The integral on the LHS does not have an elementary antiderivative, so how do one prove that this is the "most general" solution since there's a possibility of obtaining a different general solution only in terms of elementary solution using a different method?
PS:
Is there a way to prove that one has found the "most" general solution by only using a particular method (e.g., integrating factors), with the possiblity of obtaining a different general solution using a different method?
For example, using integrating factor in the ODE
y' + x²y = x
exp(x³/3) y' + exp(x³/3) x²y = exp(x³/3) x
d(exp(x³/3) y) = exp(x³/3)x dx
Integrate both sides:
[tex]exp(x^3/3) y = \int \exp(x^3/3)x\ dx[/tex] or
[tex]y = \frac{\int \exp(x^3/3)x\ dx}{\exp(x^3/3)}[/tex]
The integral on the LHS does not have an elementary antiderivative, so how do one prove that this is the "most general" solution since there's a possibility of obtaining a different general solution only in terms of elementary solution using a different method?