forked from Sasserisop/MATH201
fixed cauchy euler
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@ -5,7 +5,7 @@ We know how to solve second order equations where a, b, c are constants. Even if
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#cauchy-euler equations are equations in the form:
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#cauchy-euler equations are equations in the form:
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$$ax^2{\frac{d^2y}{dx^2}+bx{\frac{ dy }{ dx }}+cy=f(x)},\ x>0$$
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$$ax^2{\frac{d^2y}{dx^2}+bx{\frac{ dy }{ dx }}+cy=f(x)},\ x>0$$
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where $a,\ b,\ c$ are still constants and $\in \mathbb{R}$
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where $a,\ b,\ c$ are still constants and $\in \mathbb{R}$
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Note: x=0 is not interesting as the derivative terms disappear.
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Note: $x=0$ is not interesting as the derivative terms disappear.
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How to solve? There are two approaches:
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How to solve? There are two approaches:
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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.
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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.
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@ -14,12 +14,14 @@ $x=e^t\Rightarrow t=\ln x$ (x is always +)
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(do $x=-e^t$ if you need it to be negative.)
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(do $x=-e^t$ if you need it to be negative.)
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find derivatives with respect to t now. y is a function of t which is a function of x.
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find derivatives with respect to t now. y is a function of t which is a function of x.
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$\frac{dy}{dx}=\frac{dy}{dt}{\frac{dt}{dx}}=\frac{ dy }{ dt }{\frac{1}{x}}\Rightarrow \underset{ \text{Important} }{ x\frac{dy}{dx}=\frac{dy}{dt} }$
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$\frac{dy}{dx}=\frac{dy}{dt}{\frac{dt}{dx}}=\frac{ dy }{ dt }{\frac{1}{x}}\Rightarrow \underset{ \text{Important} }{ x\frac{dy}{dx}=\frac{dy}{dt} }$
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compute 2nd derivative of y wrt to x:
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using $\frac{dy}{dx}=\frac{dy}{dt}{\frac{dt}{dx}}$ and the chain rule, compute 2nd derivative of y wrt to x:
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$\frac{d^2y}{dx^2}=\frac{d^2y}{dt^2} \frac{dt}{dx}\cdot \frac{dt}{dx}+\frac{dy}{dt}{\frac{d^2t}{dx^2}}=\frac{1}{x^2}{\frac{d^2y}{dt^2}}-\frac{\frac{1}{x^2}dy}{dt}$
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$\frac{d^2y}{dx^2}=\frac{d^2y}{dt^2} \frac{dt}{dx}\cdot \frac{dt}{dx}+\frac{dy}{dt}{\frac{d^2t}{dx^2}}=\frac{1}{x^2}{\frac{d^2y}{dt^2}}-\frac{1}{x^2}\frac{dy}{dt}$
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$\underset{ \text{Important} }{ x^2{\frac{d^2y}{dx^2}}=\frac{d^2y}{dt^2}-\frac{dy}{dt} }$
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$\underset{ \text{Important} }{ x^2{\frac{d^2y}{dx^2}}=\frac{d^2y}{dt^2}-\frac{dy}{dt} }$
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plugging those derivatives in we get: #remember
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plugging those derivatives in we get: #remember
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$$a\frac{d^2y}{dt^2}+(b-a){\frac{dy}{dt}}+cy(t)=f(e^t)$$
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$$a\frac{d^2y}{dt^2}+(b-a){\frac{dy}{dt}}+cy(t)=f(e^t)$$
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^ this is a constant coefficient second order non-homogenous equation now! We can solve it now using prior tools.
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^ this is a constant coefficient second order non-homogenous equation now! We can solve it now using prior tools.
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> If you make the substitution $x=-e^t$ and go through the derivation, you get:
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> $a\frac{d^2y}{dt^2}+(b-a){\frac{dy}{dt}}+cy(t)=f(-e^t)$ <- Very nice that it's so similar, makes it easy to remember.
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## Example:
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## Example:
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#ex #second_order #second_order_nonhomogenous #cauchy-euler
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#ex #second_order #second_order_nonhomogenous #cauchy-euler
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@ -28,14 +30,17 @@ $$x^2{\frac{d^2y}{dx^2}}+3x{\frac{dy}{dx}}+y=x^{-1},\ x>0$$
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substitute: $x=e^t$
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substitute: $x=e^t$
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transform using the technique we showed just earlier:
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transform using the technique we showed just earlier:
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$\frac{d^2y}{dt^2}+2{\frac{dy}{dt}}+y=e^{-t}$
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$\frac{d^2y}{dt^2}+2{\frac{dy}{dt}}+y=e^{-t}$
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1) $r^2-2r+1=0$
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1) $r^2+2r+1=0$
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$r_{1,2}=-1$
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$r_{1,2}=-1$
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$y_{h}(t)=c_{1}e^{-t}+c_{2}te^{-t}$
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$y_{h}(t)=c_{1}e^{-t}+c_{2}te^{-t}$
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2) $y_{p}(t)=At^2e^{-t}$ <- using method of undetermined coefficients
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2) $y_{p}(t)=At^2e^{-t}$ <- using method of undetermined coefficients
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$\underbrace{ \cancel{ At^2e^{-t} }+\cancel{ A 2t(-e^{-t}) }+2Ae^{-t}\cancel{ -2Ate^{-t} } }_{ y_{p}'' }\quad+\underbrace{ \cancel{ 2At^2(-e^{-t}) }+\cancel{ 2A 2te^{-t} } }_{ 2y_{p}' }\quad+\underbrace{\cancel{ At^2e^{-t} } }_{ y_{p} }=e^{-t}$
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$2Ae^{-t}=e^{-t}$
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$A=\frac{1}{2}$
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$A=\frac{1}{2}$
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general solution in terms of t:
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general solution in terms of $t$:
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$y(t)=c_{1}e^{-t}+c_{2}te^{-t}+\frac{1}{2}t^2e^{-t}$
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$y(t)=c_{1}e^{-t}+c_{2}te^{-t}+\frac{1}{2}t^2e^{-t}$
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but we want solution in terms of x:
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but we want solution in terms of $x$:
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$y(x)=c_{1}e^{-\ln(x)}+c_{2}\ln(x)e^{-\ln(x)}+\frac{1}{2}\ln(x)^2e^{-\ln(x)}$ <- This is rather lousy notation, the y here isn't the same as the y above. Conceptually though, it's all oke doke.
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$y(x)=c_{1}e^{-\ln(x)}+c_{2}\ln(x)e^{-\ln(x)}+\frac{1}{2}\ln(x)^2e^{-\ln(x)}$ <- This is rather lousy notation, the y here isn't the same as the y above. Conceptually though, it's all oke doke.
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$$y(x)=c_{1}x^{-1}+c_{2}\ln(x)x^{-1}+\frac{1}{2}{\ln(x)^2}x^{-1}$$
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$$y(x)=c_{1}x^{-1}+c_{2}\ln(x)x^{-1}+\frac{1}{2}{\ln(x)^2}x^{-1}$$
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We are done.
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We are done.
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