Solving differential equations (circular motion)

[Gatsby88]

Homework Statement

I have a differential equation of the form

$\frac{dZ}{d\theta} + cZ = a cos \theta + b sin \theta$

Where $Z = \frac{1}{2}\dot{\theta}^{2}$

I need to find the general solution of this equation. a, b and c are all constants.

Homework Equations

The questions suggests using this to help:

$\int e^{\lambda x} (a cos x + bsin x ) = \frac{1}{1+\lambda^2}e^{\lambda x}(\lambda (a cos x + b sin x) a sin x - b cos x) + C$

The Attempt at a Solution

I just don't know how that integral is supposed to help me solve the equation. How does e become relevant to this function?

Im also a bit unsure about this.. If I integrate

$\frac{1}{2}C \dot{\theta}^2$

with respect to θ, do I get

$\frac{1}{2}C \theta ^2$ ?

 

[Curious3141]

Homework Statement

I have a differential equation of the form

$\frac{dZ}{d\theta} + cZ = a cos \theta + b sin \theta$

Where $Z = \frac{1}{2}\dot{\theta}^{2}$

I need to find the general solution of this equation. a, b and c are all constants.

Homework Equations

The questions suggests using this to help:

$\int e^{\lambda x} (a cos x + bsin x ) = \frac{1}{1+\lambda^2}e^{\lambda x}(\lambda (a cos x + b sin x) a sin x - b cos x) + C$

The Attempt at a Solution

I just don't know how that integral is supposed to help me solve the equation. How does e become relevant to this function?

The first step involves solving for $Z$ in terms of $\theta$. Are you familiar with the technique of using an integrating factor? You can read more about it here: http://en.wikipedia.org/wiki/Integrating_factor

Multiply by the appropriate integrating factor, and the reason for the hint should become very clear.

Im also a bit unsure about this.. If I integrate

$\frac{1}{2}C \dot{\theta}^2$

with respect to θ, do I get

$\frac{1}{2}C \theta ^2$ ?

No. Remember that $\dot \theta$ signifies a derivative wrt time. There's no (simple) relationship between the two expressions.

 

[Gatsby88]

Thank you.

Ah. Yes, we've done integrating factors earlier in the course.

The integrating factor will be

$e^{\int {c}} = e^{c\theta}$

multiplying by this gives

$e^{c\theta} \frac{dZ}{d\theta} + C e^{c\theta} Z = e^{c\theta}(a cos \theta + b sin \theta)$

And then

$\frac{d}{d\theta}(e^{c\theta}Z) = e^{c\theta}(a cos \theta + b sin \theta)$

And then I can integrate both sides and use the integral given as well as substituting back in for Z at the end and everything is finished.

However, I largely just followed a 'process solution' for this from my textbook. I am unsure how the LHS goes from

$e^{c\theta} \frac{dZ}{d\theta} + C e^{c\theta} Z$

to

$\frac{d}{d\theta}(e^{c\theta}Z)$

Can someone explain that to me please?