Bernoulli Equation Checking the solution

In summary, the conversation discusses solving a Bernoulli equation by finding the integrating factor and checking the solution using Mathematica code. The conversation also highlights the reliability of this method in verifying the solution.
  • #1
cbarker1
Gold Member
MHB
347
23
Dear Everyone,

I need someone to check the solution,

The question identity and then solve,

The Equation is Bernoulli
To Solve:

$(ye^{-2x}+y^3)dx-e^{-2x}dy=0$

$\frac{1}{dx}(ye^{x}+y^3)dx=\frac{1}{dx}(e^{2x}dy)$

$ye^{-2x}+y^3=e^{-2x}\d{y}{x}$

$e^{-2x}y^{\prime}-ye^{-2x}=y^3$

Let $v={y}^{1-3}$

$v={y}^{-2}\implies {v}^{-\frac{1}{2}}=y$

$-\frac{1}{2}{v}^{-\frac{3}{2}}v^{\prime}=y^{\prime}$

$-\frac{1}{2}{v}^{-\frac{3}{2}}e^{-2x}v^{\prime}-e^{-2x}{v}^{-\frac{1}{2}}={\left\{{v}^{-\frac{1}{2}}\right\}}^{3}\implies-\frac{1}{2}{v}^{-\frac{3}{2}}e^{-2x}v^{\prime}-e^{-2x}{v}^{-\frac{1}{2}}={v}^{-\frac{3}{2}}$

${v}^{\frac{3}{2}}\left\{-\frac{1}{2}{v}^{-\frac{3}{2}}e^{-2x}v^{\prime}-e^{-2x}{v}^{-\frac{1}{2}}\right\}={v}^{-\frac{3}{2}}{v}^{\frac{3}{2}}$

$e^{-2x}v^{\prime}+2e^{-2x}v=-2$

$e^{2x}\left\{e^{-2x}v^{\prime}+2e^{-2x}v\right\}=-2e^{2x}$

$v^{\prime}+2v=-2e^{2x}$
Finding the Integrating Factor:

$\mu(x)=e^{\int2dx}=e^{2x}$

$e^{2x}v^{\prime}+2e^{2x}v=-2e^{2x}e^{2x}$

$\left[e^{2x}v\right]^{\prime}=-2e^{4x}$

$\int\left[e^{2x}v\right]^{\prime}=\int-2e^{4x}dx$

$e^{2x}v=-\frac{1}{2}e^{4x}+C$

$v=-\frac{1}{2}e^{2x}+Ce^{-2x}$

${y}^{-2}=-\frac{1}{2}e^{2x}+Ce^{-2x}$
 
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  • #2
The cool thing about solving DE's is that you always have a good, easy, reliable way to check your answer: plug back into the DE. In your case, the Mathematica code
Code:
y[x_]=(-(1/2)*Exp[2x]+C[1] Exp[-2x])^(-1/2) 
Simplify[y[x] Exp[-2x]+(y[x])^3-Exp[-2x]y'[x]]
produced a zero, so I would say you certainly found a solution!
 

Related to Bernoulli Equation Checking the solution

1. What is the Bernoulli equation?

The Bernoulli equation is a fundamental equation in fluid mechanics that describes the relationship between pressure, velocity, and elevation in a fluid flow. It is based on the principle of conservation of energy and is commonly used to analyze flow in pipes, pumps, and other fluid systems.

2. How is the Bernoulli equation derived?

The Bernoulli equation is derived from the application of the conservation of energy principle to a small volume of fluid in a steady flow. This results in the equation: P + (1/2)ρv^2 + ρgh = constant, where P is pressure, ρ is density, v is velocity, g is acceleration due to gravity, and h is elevation.

3. What are the assumptions made in the Bernoulli equation?

The Bernoulli equation assumes that the fluid flow is steady, incompressible, and inviscid (no friction). It also assumes that the fluid has a constant density and that the flow is along a streamline, meaning there is no change in direction or eddies in the flow.

4. How is the Bernoulli equation used to check a solution?

The Bernoulli equation can be used to check a solution by plugging in the known values of pressure, velocity, and elevation at two different points in a fluid system. The resulting equation should have the same value on both sides, indicating that the solution is valid.

5. What are some common applications of the Bernoulli equation?

The Bernoulli equation has many practical applications, including analyzing the lift of an airplane wing, calculating the flow rate of a fluid through a pipe, and determining the efficiency of a pump. It is also used in the design of hydraulic systems, such as dams and water turbines.

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