fixed cauchy euler

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Sasserisop 2023-12-20 20:51:20 -07:00
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@ -5,7 +5,7 @@ We know how to solve second order equations where a, b, c are constants. Even if
#cauchy-euler equations are equations in the form: #cauchy-euler equations are equations in the form:
$$ax^2{\frac{d^2y}{dx^2}+bx{\frac{ dy }{ dx }}+cy=f(x)},\ x>0$$ $$ax^2{\frac{d^2y}{dx^2}+bx{\frac{ dy }{ dx }}+cy=f(x)},\ x>0$$
where $a,\ b,\ c$ are still constants and $\in \mathbb{R}$ where $a,\ b,\ c$ are still constants and $\in \mathbb{R}$
Note: x=0 is not interesting as the derivative terms disappear. Note: $x=0$ is not interesting as the derivative terms disappear.
How to solve? There are two approaches: How to solve? There are two approaches:
Textbook only use 2nd method, prof doesn't like this. You can find both methods in the profs notes. Btw, do you know Stewart? Multimillionaire, he's living in a mansion in Ontario. Textbook only use 2nd method, prof doesn't like this. You can find both methods in the profs notes. Btw, do you know Stewart? Multimillionaire, he's living in a mansion in Ontario.
@ -14,12 +14,14 @@ $x=e^t\Rightarrow t=\ln x$ (x is always +)
(do $x=-e^t$ if you need it to be negative.) (do $x=-e^t$ if you need it to be negative.)
find derivatives with respect to t now. y is a function of t which is a function of x. find derivatives with respect to t now. y is a function of t which is a function of x.
$\frac{dy}{dx}=\frac{dy}{dt}{\frac{dt}{dx}}=\frac{ dy }{ dt }{\frac{1}{x}}\Rightarrow \underset{ \text{Important} }{ x\frac{dy}{dx}=\frac{dy}{dt} }$ $\frac{dy}{dx}=\frac{dy}{dt}{\frac{dt}{dx}}=\frac{ dy }{ dt }{\frac{1}{x}}\Rightarrow \underset{ \text{Important} }{ x\frac{dy}{dx}=\frac{dy}{dt} }$
compute 2nd derivative of y wrt to x: using $\frac{dy}{dx}=\frac{dy}{dt}{\frac{dt}{dx}}$ and the chain rule, compute 2nd derivative of y wrt to x:
$\frac{d^2y}{dx^2}=\frac{d^2y}{dt^2} \frac{dt}{dx}\cdot \frac{dt}{dx}+\frac{dy}{dt}{\frac{d^2t}{dx^2}}=\frac{1}{x^2}{\frac{d^2y}{dt^2}}-\frac{\frac{1}{x^2}dy}{dt}$ $\frac{d^2y}{dx^2}=\frac{d^2y}{dt^2} \frac{dt}{dx}\cdot \frac{dt}{dx}+\frac{dy}{dt}{\frac{d^2t}{dx^2}}=\frac{1}{x^2}{\frac{d^2y}{dt^2}}-\frac{1}{x^2}\frac{dy}{dt}$
$\underset{ \text{Important} }{ x^2{\frac{d^2y}{dx^2}}=\frac{d^2y}{dt^2}-\frac{dy}{dt} }$ $\underset{ \text{Important} }{ x^2{\frac{d^2y}{dx^2}}=\frac{d^2y}{dt^2}-\frac{dy}{dt} }$
plugging those derivatives in we get: #remember plugging those derivatives in we get: #remember
$$a\frac{d^2y}{dt^2}+(b-a){\frac{dy}{dt}}+cy(t)=f(e^t)$$ $$a\frac{d^2y}{dt^2}+(b-a){\frac{dy}{dt}}+cy(t)=f(e^t)$$
^ this is a constant coefficient second order non-homogenous equation now! We can solve it now using prior tools. ^ this is a constant coefficient second order non-homogenous equation now! We can solve it now using prior tools.
> If you make the substitution $x=-e^t$ and go through the derivation, you get:
> $a\frac{d^2y}{dt^2}+(b-a){\frac{dy}{dt}}+cy(t)=f(-e^t)$ <- Very nice that it's so similar, makes it easy to remember.
## Example: ## Example:
#ex #second_order #second_order_nonhomogenous #cauchy-euler #ex #second_order #second_order_nonhomogenous #cauchy-euler
@ -28,14 +30,17 @@ $$x^2{\frac{d^2y}{dx^2}}+3x{\frac{dy}{dx}}+y=x^{-1},\ x>0$$
substitute: $x=e^t$ substitute: $x=e^t$
transform using the technique we showed just earlier: transform using the technique we showed just earlier:
$\frac{d^2y}{dt^2}+2{\frac{dy}{dt}}+y=e^{-t}$ $\frac{d^2y}{dt^2}+2{\frac{dy}{dt}}+y=e^{-t}$
1) $r^2-2r+1=0$ 1) $r^2+2r+1=0$
$r_{1,2}=-1$ $r_{1,2}=-1$
$y_{h}(t)=c_{1}e^{-t}+c_{2}te^{-t}$ $y_{h}(t)=c_{1}e^{-t}+c_{2}te^{-t}$
2) $y_{p}(t)=At^2e^{-t}$ <- using method of undetermined coefficients 2) $y_{p}(t)=At^2e^{-t}$ <- using method of undetermined coefficients
$\underbrace{ \cancel{ At^2e^{-t} }+\cancel{ A 2t(-e^{-t}) }+2Ae^{-t}\cancel{ -2Ate^{-t} } }_{ y_{p}'' }\quad+\underbrace{ \cancel{ 2At^2(-e^{-t}) }+\cancel{ 2A 2te^{-t} } }_{ 2y_{p}' }\quad+\underbrace{\cancel{ At^2e^{-t} } }_{ y_{p} }=e^{-t}$
$2Ae^{-t}=e^{-t}$
$A=\frac{1}{2}$ $A=\frac{1}{2}$
general solution in terms of t: general solution in terms of $t$:
$y(t)=c_{1}e^{-t}+c_{2}te^{-t}+\frac{1}{2}t^2e^{-t}$ $y(t)=c_{1}e^{-t}+c_{2}te^{-t}+\frac{1}{2}t^2e^{-t}$
but we want solution in terms of x: but we want solution in terms of $x$:
$y(x)=c_{1}e^{-\ln(x)}+c_{2}\ln(x)e^{-\ln(x)}+\frac{1}{2}\ln(x)^2e^{-\ln(x)}$ <- This is rather lousy notation, the y here isn't the same as the y above. Conceptually though, it's all oke doke. $y(x)=c_{1}e^{-\ln(x)}+c_{2}\ln(x)e^{-\ln(x)}+\frac{1}{2}\ln(x)^2e^{-\ln(x)}$ <- This is rather lousy notation, the y here isn't the same as the y above. Conceptually though, it's all oke doke.
$$y(x)=c_{1}x^{-1}+c_{2}\ln(x)x^{-1}+\frac{1}{2}{\ln(x)^2}x^{-1}$$ $$y(x)=c_{1}x^{-1}+c_{2}\ln(x)x^{-1}+\frac{1}{2}{\ln(x)^2}x^{-1}$$
We are done. We are done.