LINEAR ODE

[Shashank Dwivedi]

Source: Linear Ordinary Differential Equation, book authored by E.A. Coddington

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I am unable to show that [tex]\varphi[/tex] is a solution of second linear homogenous differential equation.
If I put directly [tex]\varphi[/tex] in the second equation and try to solve, I am coming across a complicated expression which doesn't seem to be zero for it to be solution.
Please someone help me in proving this.

[HallsofIvy]

You aren't asked to show that [tex]\phi[/tex] is a solution to a differential equation! You are told that [tex]\phi[/tex] is a solution to the first equation and asked to show that [tex]\psi(x)= \phi(x)e^{ax/n}[/tex] is a solution to the second equation. I would suggest that you work out some derivatives of [tex]\psi[/tex] in terms of the derivatives of [tex]\phi[/tex]. With [tex]\psi(x)= \phi(x)e^{ax/n}[/tex], [tex]\psi'(x)= \phi'(x)^{ax/n}+ \frac{a}{n}\phi(x)e^{ax/n}= (\phi(x)+ \frac{a}{n}\phi'(x))e^{ax/n}[/tex], [tex]\psi''(x)= (\phi''(x)+ \frac{2a}{n}\phi'(x)+ \frac{a^2}{n^2}\phi)e^{ax/n}[/tex], etc. (You might be reminded of binomial coefficients.)