Violating intial conditions: ODEs

[CharlesNguyen]

Hi Everyone,

I had a quick question. If you have an IVP ODE and you solve for the general solution first and you had fractions in it, could you multiply by a number to make it "easier" (whole number, rather than involving fractions) without violating the initial conditions?

Thanks

 

[Geofleur]

It depends on whether the initial conditions are homogeneous or not. For example, if you had something like $x(0) = 0$ and $\frac{dx}{dt}(0) = 0$, you could multiply the solution to the ODE by any nonzero number you want, and it would still satisfy the ICs. But if you had, say $x(0) = 1$, you would have a problem.

 

[bigfooted]

In general, you need to transform your initial conditions as well, for example,

$y'=x$, $y(0)=1$

has as general solution $y=\frac{1}{2}x^2 + C$
and C=1 when you use the initial condition.
You can multiply the general solution by 2 and use the transformation z=2y to get rid of the fractions:
$z=x^2 + 2C$
but you also have to multiply the original ODE by 2 (because dz/dx=2dy/dx) and the initial condition to rewrite it to z:
$z'=2x$, $z(0)=2$