# Thread: second order ODE

1. ## second order ODE

Solving v''t + v' = 0

When I substitute w for v' I get

$\frac{dw}{w}=-\frac{dt}{t}$

$ln(t)=-ln(w)+c$

but this gives me $w=-t/c \rightarrow w=t/c$ rather than $w=c/t$

A seemingly simple separable equation that I am getting hung up on.

2. Originally Posted by petition Edgecombe
Solving v''t + v' = 0

When I substitute w for v' I get

$\frac{dw}{w}=-\frac{dt}{t}$

$ln(t)=-ln(w)+c$

but this gives me $w=-t/c \rightarrow w=t/c$ rather than $w=c/t$

A seemingly simple separable equation that I am getting hung up on.
how did you get that?

write c as ln(C), then what happens?

$\ln t = - \ln w + \ln C$

but $\ln X - \ln Y = \ln (X/Y)$, so that

$\ln t = \ln (C/w)$

$\Rightarrow t = \frac Cw \implies w = \frac Ct$