Im sorry, I posted this in the wrong section, feel free to move it to the homework section.(adsbygoogle = window.adsbygoogle || []).push({});

Hey guys, Ive really been needing some help with this one. Im doing an assignment for Ordinary Differential Equations and I was hoping someone could help me out by looking over my work. Ive been working on this assignment for a couple of weeks now and finally decided that I just needed help.

There is a second part of this problem, but Ill post that later depending on whether or not this part is correct.

Anyway, I know this is a really long post but I will truly appreciate the help of anyone thats willing to go through it all.

Basically I just need to solve the general equation below for different values of gamma and lambda.

Thanks

[tex]

General Equation

[/tex]

[tex]

\frac {dm}{dt}=-\alpha m^{\gamma}-\lambda m

[/tex]

[tex]

\lambda = constant[/tex]

[tex]

Case (1)

[/tex]

[tex] \gamma=1[/tex]

1a.) [tex]\frac{dm}{dt}=-\alpha(m)-\lambda(m) [/tex]

2a.) [tex]\frac{dm}{dt}=-m(\alpha+\lambda) [/tex]

3a.) [tex]\int \frac{1}{m}dm=\int (\alpha+\lambda)dt[/tex]

4a.) [tex]\ln(m)=-(\alpha+\lambda)t+C[/tex]

5a.) [tex]m=e^{-(\alpha+\lambda)t+C}[/tex]

[tex]Case (2) [/tex]

constraints

[tex]\gamma cannot =0[/tex] [tex]\lambda=0[/tex]

Simplifying the general equation I get

1b.) [tex]\frac{dm}{dt}= -\alpha (m)^\gamma[/tex]

2b.) [tex]\int \frac{1}{m^\gamma}dm=-\int \alpha dt[/tex]

3b.) [tex]\frac{m^{(-\gamma + 1)}}{-\gamma+1}=-\alpha (t)+C[/tex]

4b.) [tex]m=[(-\alpha (t)+C)(-\gamma+1)]^{1/(-\gamma+1)} [/tex]

[tex]Case (3) [/tex]

Constraints

[tex]\gamma not=1 \lambda not =0[/tex]

1c.) [tex]\frac{dm}{dt}=-\alpha (m^{\gamma)}-\lambda (m) [/tex]

I then solved this using the Bernoulli method

2c.) [tex]v\equiv m^{(1-\gamma)} for \gamma cannot =1[/tex]

3c.) [tex]\frac{dv}{dt}=(1-\gamma)m^{(-\gamma)} \frac{dm}{dt}[/tex]

4c.) [tex]\frac{dm}{dt}+\lambda (m) =-\alpha (m^{\gamma)} [/tex]

5c.) [tex]m^{(-\gamma)}\frac{dm}{dt}=-\alpha-\lambda m^{(1-\gamma)} [/tex]

6c.) [tex]m^{(-\gamma)} \frac{dm}{dt}=-\alpha-\lambda (v) [/tex]

Substituting dv/dt into the equation I get

7c.) [tex]\frac{dv}{dt}=(1-\gamma) m^{-\gamma} m^{\gamma}(-\alpha-\lambda v) [/tex]

8c.) [tex]\frac{dv}{dt}=(1-\gamma)(-\alpha-\lambda v) [/tex]

9c.) [tex]\frac{dv}{dt}+(1-\gamma)(\lambda)(v)= -(1-\gamma)\alpha [/tex]

This is a linear first order homogeneous equation and can be solved by making

10c.) [tex]\mu=e^{\int(1-\gamma)\lambda dt} =e^{(1-\gamma)\lambda(t)} [/tex]

using this as my integrating factor I get this

11c.) [tex]v=\frac{1}{e^{(1-\gamma(t)}}\int e^{(1-\gamma)t} (1-\gamma) (-\alpha)dt[/tex]

this yields

12c.) [tex]v=-\alpha +\frac{C}{e^{((1-\gamma)t)}} [/tex]

substituting for v we get

13c.) [tex]m=(-\alpha+ \frac{C}{e^{((1-\gamma)t))}}^{(\frac{1}{1-\gamma})}

[/tex]

Am I doing anything wrong?

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# Homework Help: ODE project please help

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