Integration and hyperbolic function problem

In summary: We make the substitution,$$u^2=a^3(\frac{\beta_0}{\Gamma_0})$$$$da=\frac{2}{3}(\frac{\Gamma_0}{\beta_0})^{\frac{1}{3}}\frac{du}{u^{\frac{1}{3}}}$$thus,$$\frac{2}{3}\frac{1}{\sqrt {\beta_0}}\int \frac{du}{\sqrt {1+u^2}}=t$$$$\sinh^{-1}(u)=\frac{
  • #1
JD_PM
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Homework Statement
While studying Cosmology I came across a particular differential equation I do not see how to solve. Please read below for details
Relevant Equations
N/A
This question arose while studying Cosmology (section 38.2 in Lecture Notes in GR) but it is purely mathematical, that is why I ask it here.

I do not see why the equation

$$H^2 = H_0^2 \left[\left( \frac{a_0}{a}\right)^3 (\Omega_M)_0 + (\Omega_{\Lambda})_0 \right] \tag{1}$$

Has the following solution

$$a(t) = a_0 \left( \frac{(\Omega_M)_0}{(\Omega_{\Lambda})_0}\right)^{1/3} \left(\sinh \left[(3/2)\sqrt{(\Omega_{\Lambda})_0}H_0t\right]\right)^{2/3}$$

Where (and ##\dot a## represents the derivative of ##a## wrt time)

$$H:=\left( \frac{\dot a}{a}\right)^2, \ \ \ \ H_0:=\left( \frac{\dot a_0}{a_0}\right)^2$$I found a similar problem here. I suspect that, to start off, we should find a smart change of variables but which one?

Might you please guide me towards the solution?

Any help is appreciated.

Thank you :biggrin:
 
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  • #2
Perhaps I can offer a suggestion you may find useful as you study DEs: They're difficult in and of themselves so sometimes it is helpful to strip them of unnecessary trappings and just study the underlying pure, clean and simpler-looking equation. With that in mind, is your equation basically:

$$\left(\frac{y'}{y}\right)^4=d\left[b\left(\frac{a}{y}\right)^3+c\right]$$
for ##a,b,c,d## some constants?
 
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  • #3
aheight said:
With that in mind, is your equation basically:

$$\left(\frac{y'}{y}\right)^4=d\left[b\left(\frac{a}{y}\right)^3+c\right]$$
for ##a,b,c,d## some constants?

Alright, I realized I made a typo here

JD_PM said:
$$H:=\left( \frac{\dot a}{a}\right)^2, \ \ \ \ H_0:=\left( \frac{\dot a_0}{a_0}\right)^2$$

It should be

$$H:=\left( \frac{\dot a}{a}\right), \ \ \ \ H_0:=\left( \frac{\dot a_0}{a_0}\right)$$So the DE we want to solve is

$$\left(\frac{y'}{y}\right)^2=d\left[b\left(\frac{a}{y}\right)^3+c\right]$$

OK at this point.

Should we start off by introducing a change of variables? Maybe it is a good idea (just guessing) to rewrite it a bit, by multiplying by ##y^2## both sides

$$y\left(y'\right)^2=d\left[ba^3+cy^3\right]$$
 
  • #4
JD_PM said:
So the DE we want to solve is

$$\left(\frac{y'}{y}\right)^2=d\left[b\left(\frac{a}{y}\right)^3+c\right]$$

Isn't that then a separable equation?
 
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  • #5
I have the following suggestions. First, for economy of notation, define the constants,
$$
\Gamma_0=H_0^2a_0^3(\Omega_{M})_0
$$
$$
\beta_0=H_0^2(\Omega_{\Lambda})_0
$$
We then have the DE,
$$
(\frac{\dot a}{a})^2=\frac{1}{a^3}[\Gamma_0 + a^3\beta_0]
$$
$$
\dot a=\frac{1}{\sqrt a}[\Gamma_0 + a^3\beta_0]^{\frac{1}{2}}
$$
which leads to,
$$
\frac{1}{\sqrt{\Gamma_0}}\int \frac{\sqrt a da}{\sqrt {1 + a^3(\frac{\beta_0}{\Gamma_0}})}=t
$$
We make the substitution,
$$
u^2=a^3(\frac{\beta_0}{\Gamma_0})
$$
$$
da=\frac{2}{3}(\frac{\Gamma_0}{\beta_0})^{\frac{1}{3}}\frac{du}{u^{\frac{1}{3}}}
$$
thus,
$$
\frac{2}{3}\frac{1}{\sqrt {\beta_0}}\int \frac{du}{\sqrt {1+u^2}}=t
$$
$$
\sinh^{-1}(u)=\frac{3}{2}\sqrt{\beta_0}t
$$
$$
u=\sinh (\frac{3}{2}\sqrt{\beta_0}t)
$$
$$
a^{\frac{3}{2}}(\frac{\beta_0}{\Gamma_0})^{\frac{1}{2}}=\sinh (\frac{3}{2}\sqrt{\beta_0}t)
$$
$$
a=(\frac{\Gamma_0}{\beta_0})^{\frac{1}{3}}[\sinh (\frac{3}{2}\sqrt{\beta_0}t)]^{\frac{2}{3}}

$$
 
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  • #6
@aheight @Fred Wright thank you very much for your help.

aheight said:
Isn't that then a separable equation?

Yes (let me use ##e## as constant instead ;))

$$\frac{y'}{y}=\left(e\left[b\left(\frac{a}{y}\right)^3+c\right]\right)^{1/2} \Rightarrow \frac{dy}{y\left(e\left[b\left(\frac{a}{y}\right)^3+c\right]\right)^{1/2}} = dt$$
 

Related to Integration and hyperbolic function problem

What are integration and hyperbolic functions?

Integration is a mathematical process that involves finding the area under a curve or the inverse of differentiation. Hyperbolic functions are a set of mathematical functions that are related to the hyperbola and are used in many scientific and engineering applications.

What are some common integration techniques?

Some common integration techniques include substitution, integration by parts, partial fractions, and trigonometric substitution. Each technique has its own advantages and is used to solve different types of integration problems.

What is the difference between integration and differentiation?

Integration and differentiation are inverse operations. While differentiation finds the rate of change of a function, integration finds the original function given its rate of change. In other words, integration is the reverse process of differentiation.

What are some real-world applications of integration and hyperbolic functions?

Integration and hyperbolic functions have various applications in fields such as physics, engineering, and economics. They are used to calculate areas, volumes, and rates of change in real-world problems, such as finding the area under a velocity-time graph or calculating the growth rate of a population.

How can I improve my skills in solving integration and hyperbolic function problems?

The best way to improve your skills in solving integration and hyperbolic function problems is through practice. Start with simple problems and gradually move on to more complex ones. You can also seek help from textbooks, online resources, or a tutor to guide you through the process.

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