- #1
Jonter
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Differential equation - distance needed to achieve target speed
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In summary, the given ODE can be rewritten as a Bernoulli equation by multiplying both sides by u. Then, by setting v=u^2, we can convert it into a linear equation. To integrate the equation, we can use the substitution v=A+Bu^2 and solve for u. The final solution is given by v=C''e^{2Bx/m}, where C'' is a constant.
- #2
MarkFL
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I would begin by writing the given ODE in the form:
\(\displaystyle \d{u}{x}-\frac{B}{m}u=\frac{A}{m}u^{-1}\)
We see we have a Bernoulli equation. Multiply by \(u\):
\(\displaystyle u\d{u}{x}-\frac{B}{m}u^2=\frac{A}{m}\)
Let \(v=u^2\) hence \(\displaystyle \d{v}{x}=2u\d{u}{x}\) and so we have:
\(\displaystyle \d{v}{x}-\frac{B}{2m}v=\frac{A}{2m}\)
We now have a linear equation, and so can you proceed?
Note: I have moved this thread to our "Differential Equations" forum.
\(\displaystyle \d{u}{x}-\frac{B}{m}u=\frac{A}{m}u^{-1}\)
We see we have a Bernoulli equation. Multiply by \(u\):
\(\displaystyle u\d{u}{x}-\frac{B}{m}u^2=\frac{A}{m}\)
Let \(v=u^2\) hence \(\displaystyle \d{v}{x}=2u\d{u}{x}\) and so we have:
\(\displaystyle \d{v}{x}-\frac{B}{2m}v=\frac{A}{2m}\)
We now have a linear equation, and so can you proceed?
Note: I have moved this thread to our "Differential Equations" forum.
- #3
HallsofIvy
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The equation \(\displaystyle mu\frac{du}{dx}= A+ Bu^2\) can be written as \(\displaystyle \frac{mu du}{A+ Bu^2}= dx\). To integrate the left side, let \(\displaystyle v= A+ Bu^2\) so that \(\displaystyle dv= 2Bu du\) or \(\displaystyle udu= \frac{dv}{2B}\). Then \(\displaystyle \frac{mu du}{A+ Bu^2}= \frac{m dv}{2Bv}= dx\). Integrating, \(\displaystyle \frac{m}{2B} ln(v)= x+ C\) or \(\displaystyle ln(v)= \frac{2Bx}{m}+ C'\) (where C'= 2BC/m) and then \(\displaystyle v= A+ Bu^2= C''e^{2Bx/m}\) (where \(\displaystyle C''= e^{C'}\)).
FAQ: Differential equation - distance needed to achieve target speed
What is a differential equation?
A differential equation is a mathematical equation that describes the relationship between a function and its derivatives. It is used to model many real-world phenomena, such as motion, growth, and decay.
How is a differential equation used to calculate the distance needed to achieve a target speed?
A differential equation can be used to model the motion of an object, including its speed and position. By setting the desired target speed as a boundary condition, the equation can be solved to determine the distance needed to achieve that speed.
What factors are involved in the differential equation for calculating distance needed to achieve target speed?
The factors involved in the differential equation include the initial velocity of the object, the acceleration, and the target speed. Other factors such as air resistance and friction may also be considered in more complex equations.
Can a differential equation be used for any type of motion?
Yes, a differential equation can be used to model any type of motion as long as the necessary factors and boundary conditions are included in the equation. This includes linear motion, circular motion, and even more complex motions such as projectile motion.
Are there any limitations to using a differential equation to calculate distance for achieving target speed?
While a differential equation can provide a mathematical solution for calculating distance needed to achieve target speed, it may not always accurately reflect real-world scenarios. Factors such as external forces, human error, and changing conditions can affect the actual distance needed to achieve a target speed.