I'm trying to solve a second order ODE in the form of Ay''+By'+1=0. This is part of a genetics problem I have, but I haven't taken DiffEq in a very long time and can't quite remember how to go about solving this. Can anyone help?
Thanks!
I'm trying to solve a second order ODE in the form of Ay''+By'+1=0. This is part of a genetics problem I have, but I haven't taken DiffEq in a very long time and can't quite remember how to go about solving this. Can anyone help?
Thanks!
I would write the ODE as follows:
$\displaystyle Ay''+By'=-1$
The characteristic roots are:
$\displaystyle 0,-\frac{B}{A}$ and so the homogeneous solution is:
$\displaystyle y_h(x)=c_1+c_2\frac{A}{B}e^{-\frac{B}{A}x}$
I would next assume a particular solution of the form:
$\displaystyle y_p(x)=kx$
Now, we may use the method of undetermined coefficients to find $\displaystyle k$:
$\displaystyle y_p'(x)=k$
$\displaystyle y_p''(x)=0$ and so:
$\displaystyle Bk=-1\,\therefore\,k=-\frac{1}{B}$ and we have:
$\displaystyle y_p(x)=-\frac{1}{B}x$
By superposition, we now may write the general solution:
$\displaystyle y(x)=y_h(x)+y_p(x)=c_1+c_2\frac{A}{B}e^{-\frac{B}{A}x}-\frac{x}{B}$