# Thread: differential equations help second order non homo

2. ## Re: differential equations help second order non homo

Have you tried what they have suggested?

3. ## Re: differential equations help second order non homo

yes but the lecturer just copied the correction from a page and skip a lot of step,so it is not making sense at all and

4. ## Re: differential equations help second order non homo

Originally Posted by Musawwir
yes but the lecturer just copied the correction from a page and skip a lot of step,so it is not making sense at all and

a) $x=e^t,~z(t)=y(x(t))$

I will use upper dot notation to denote time derivatives and prime notation to denote derivatives of $y$ with respect to $x$ I.e.

$\dot{z} = \dfrac{dz}{dt}$

$y^\prime = \dfrac{dy}{dx}$

$\dot{z} = y^\prime \dot{x} = y^\prime e^t$

$y^\prime = \dot{z} e^{-t}$

\begin{align*} &\ddot{z}= y^{\prime\prime}(\dot{x}^2) + y^\prime \dot{x} \\ &=y^{\prime\prime} e^{2t} + y^\prime e^t \\ &=y^{\prime\prime} e^{2t} + \dot{z} \end{align*}

$y^{\prime\prime} = (\ddot{z}-\dot{z})e^{-2t}$

Now you plug $y^\prime, ~y^{\prime\prime}$ into the original differential equation, letting $x=e^t$, and solve it in $t$

Finally let $t = \ln(x)$

5. ## Re: differential equations help second order non homo

this is the only line i was not understanding z¨=y′′(x˙2)+y′x

6. ## Re: differential equations help second order non homo

Originally Posted by Musawwir
this is the only line i was not understanding z¨=y′′(x˙2)+y′x
$\dot{z} = y^\prime \dot{x}$

\begin{align*} &\dfrac{d}{dt} \dot{z} = \\ &\left(\dfrac{d}{dt} y^\prime\right)\dot{x} + y^\prime \ddot{x} = \\ &\left(\dfrac{d}{dx} y^\prime \dot{x}\right)\dot{x} + y^\prime \ddot{x} = \\ &\left(y^{\prime\prime}\dot{x}\right)\dot{x} + y^\prime \ddot{x} = \\ &y^{\prime\prime}\dot{x}^2 + y^\prime \ddot{x} \end{align*}