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First order of equation: (dy/dx) + (cot x)y = csc x
- Thread starter warsatan
- Start date
This actually falls in place pretty easily.
You have a linear first order DE.
\(\displaystyle \L\\\frac{dy}{dx}+ycot(x)=csc(x)\)
Find the integrating factor:
\(\displaystyle \L\\e^{\int{cot(x)}dx}=e^{ln(sin(x))}=sin(x)\)
\(\displaystyle \L\\\frac{d}{dx}[ysin(x)]=sin(x)csc(x)\)
Integrate:
\(\displaystyle \L\\ysin(x)=x+C\)
Solve for y:
\(\displaystyle \L\\y=\frac{x+C}{sin(x)}\)
You have a linear first order DE.
\(\displaystyle \L\\\frac{dy}{dx}+ycot(x)=csc(x)\)
Find the integrating factor:
\(\displaystyle \L\\e^{\int{cot(x)}dx}=e^{ln(sin(x))}=sin(x)\)
\(\displaystyle \L\\\frac{d}{dx}[ysin(x)]=sin(x)csc(x)\)
Integrate:
\(\displaystyle \L\\ysin(x)=x+C\)
Solve for y:
\(\displaystyle \L\\y=\frac{x+C}{sin(x)}\)
okay, working on the last one tonight, here what i have so far
y' +ay = be ^-kt
\(\displaystyle \L\\\frac{dy}{dx}+ay=be^{-kt}\)
\(\displaystyle \L\\e^{\int{a}dt}=e^{t}\)
\(\displaystyle \L\\\frac{d}{dx}[ye^{t}]=b e^{t} e^{-kt]\)
how's that look so far? what i'm i gonna do with those two e on the right side? btw, -k = - lamda ( i dont know how to get the symbol). Thanks
y' +ay = be ^-kt
\(\displaystyle \L\\\frac{dy}{dx}+ay=be^{-kt}\)
\(\displaystyle \L\\e^{\int{a}dt}=e^{t}\)
\(\displaystyle \L\\\frac{d}{dx}[ye^{t}]=b e^{t} e^{-kt]\)
how's that look so far? what i'm i gonna do with those two e on the right side? btw, -k = - lamda ( i dont know how to get the symbol). Thanks
Your integrating factor is wrong. It should be \(\displaystyle e^{at}\)
Skipping ahead to the end after integration
\(\displaystyle \L\\ye^{at}=\frac{be^{(a-\lambda)t}}{a-k}+C\)
Divide through to isolate y:
\(\displaystyle \L\\y=\frac{be^{{-\lambda}t}}{a-k}+Ce^{-at}\)
Incidentally, to get a Greek letter, type {\lambda}. If you want a capital just capitalize the first letter.
Skipping ahead to the end after integration
\(\displaystyle \L\\ye^{at}=\frac{be^{(a-\lambda)t}}{a-k}+C\)
Divide through to isolate y:
\(\displaystyle \L\\y=\frac{be^{{-\lambda}t}}{a-k}+Ce^{-at}\)
Incidentally, to get a Greek letter, type {\lambda}. If you want a capital just capitalize the first letter.