- #1
ManishR
- 88
- 0
Consider
[tex]\frac{d^{2}y}{dx^{2}}=k[/tex]--------------[0]
and
[tex]\frac{dy}{dx}=y_{a}[/tex]
Then
[tex]y_{f}-y_{i}=\frac{k}{2}(x_{f}^{2}-x_{i}^{2})[/tex]-----------[1]
[tex]y_{af}-y_{ai}=k(x_{f}-x_{i})[/tex]-------------[2]
is equation 1 and 2 correct ? if no, then what is the correct solution
if yes
[tex]\Rightarrow\frac{d^{2}y}{dx^{2}}=\frac{dy}{dx}=(positive/negative)[/tex]
or more correctly
[tex]\Rightarrow\frac{\frac{d^{2}y}{dx^{2}}}{\left|\frac{d^{2}y}{dx^{2}}\right|}=\frac{\frac{dy}{dx}}{\left|\frac{dy}{dx}\right|}=\frac{k}{\left|k\right|}[/tex]
assumming
[tex]x_{f}>x_{i}[/tex]
but that's not always true for example its not true for this
[tex]y=2x-x^{2}[/tex]-------------------[4]
where
[tex]-1\leq x\leq1[/tex]
[tex]\Rightarrow\frac{dy}{dx}=2-2x=positive[/tex]
[tex]\Rightarrow\frac{d^{2}y}{dx^{2}}=-2=negative[/tex]
so how to derive from integral equation (like [0]) to normal equation(like [4]) ?
[tex]\frac{d^{2}y}{dx^{2}}=k[/tex]--------------[0]
and
[tex]\frac{dy}{dx}=y_{a}[/tex]
Then
[tex]y_{f}-y_{i}=\frac{k}{2}(x_{f}^{2}-x_{i}^{2})[/tex]-----------[1]
[tex]y_{af}-y_{ai}=k(x_{f}-x_{i})[/tex]-------------[2]
is equation 1 and 2 correct ? if no, then what is the correct solution
if yes
[tex]\Rightarrow\frac{d^{2}y}{dx^{2}}=\frac{dy}{dx}=(positive/negative)[/tex]
or more correctly
[tex]\Rightarrow\frac{\frac{d^{2}y}{dx^{2}}}{\left|\frac{d^{2}y}{dx^{2}}\right|}=\frac{\frac{dy}{dx}}{\left|\frac{dy}{dx}\right|}=\frac{k}{\left|k\right|}[/tex]
assumming
[tex]x_{f}>x_{i}[/tex]
but that's not always true for example its not true for this
[tex]y=2x-x^{2}[/tex]-------------------[4]
where
[tex]-1\leq x\leq1[/tex]
[tex]\Rightarrow\frac{dy}{dx}=2-2x=positive[/tex]
[tex]\Rightarrow\frac{d^{2}y}{dx^{2}}=-2=negative[/tex]
so how to derive from integral equation (like [0]) to normal equation(like [4]) ?