Bernoulli Equation with weird integral

In summary, the conversation is discussing the possibility of using a u-substitution or integration by parts to solve the equation z'+Pz=Q. The speaker mentions a potential mistake in the integration process and suggests a change of variable to simplify the integration of the right hand side.
  • #1
acalcstudent
1
0
TL;DR Summary
Differential equation ending with an integral that doesn't make sense
q2.png


Part of me thinks this is could be a u-sub b/c x^3's derivative is 3x^2, a factor of 3 off from what e is raised to...but it is not a traditional u-sub...any thoughts if this is a u-sub or by parts, and what u should be?I know that there is more to solving the equation after this ( z = e^{1/(x^2)}(c_1+[insert integral from above], y = z^2) but i can't get to that without the integral above.
 
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  • #2
I presume you're trying to solve
[tex]z'+Pz=Q[/tex]
I think you made a mistake:
[tex]-\int p(x)dx=-x^{2}[/tex] and not what you wrote down which was [itex]x^{-2}[/itex]. you could have cleverly spotted that multiplying throughout by [itex]e^{x^{2}}[/itex] the LHS is a derivative. Integrating the RHS can be written (with the appropriate change of variable (which you can find for yourself)):
[tex]\frac{1}{2}\int ue^{u}du[/tex]
 

Related to Bernoulli Equation with weird integral

1. What is the Bernoulli Equation with weird integral?

The Bernoulli Equation with weird integral is a mathematical formula used to describe the relationship between pressure, velocity, and height in a fluid flow. It is an extension of the original Bernoulli Equation, which includes an additional term involving an integral.

2. How is the Bernoulli Equation with weird integral derived?

The Bernoulli Equation with weird integral is derived from the Navier-Stokes equations, which describe the motion of fluids. It is a result of applying the principles of conservation of energy along a streamline in a fluid flow.

3. What is the significance of the weird integral in the Bernoulli Equation?

The weird integral in the Bernoulli Equation represents the work done by the pressure forces on the fluid as it flows along a streamline. It takes into account the changes in pressure along the streamline, making the equation more accurate for real-world fluid flows.

4. Can the Bernoulli Equation with weird integral be used for all types of fluid flows?

Yes, the Bernoulli Equation with weird integral can be used for all types of fluid flows, including incompressible and compressible fluids. However, it is most accurate for steady, incompressible flows with negligible viscous effects.

5. How is the Bernoulli Equation with weird integral applied in practical situations?

The Bernoulli Equation with weird integral is commonly used in fluid mechanics and aerodynamics to analyze and design various systems, such as pipes, pumps, and airfoils. It is also used in the study of weather patterns and in the design of aircraft and other vehicles.

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