DE Identifying/Solving Homo/Bernoullis/Exact/etc

  • Thread starter netapparition
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In summary, DE Identifying is the process of determining the type and order of a differential equation, solved through techniques such as integration and substitution. In a scientific context, Homo is solved by studying evolutionary history, genetics, and physical characteristics. Bernoulli's equation is significant in physics and is solved using principles of conservation and mathematical techniques. Exact and inexact differential equations differ in their solvability and properties. Scientists use differential equations to model and solve complex problems in various fields.
  • #1
netapparition
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Can anyone confirm that I have identified the proper types of Differential Equations and if anyone has the time to write out a decent notation of how to solve the following sampel questions I would appreciate it. We are using the Elementary Differential Equations Eigth Edition book by Boyle and DiPrima which has inconsistant notation.

a. (dy/dx)+y=y^2(cos(x)-sin(x)) => Bernoulli equation

b. (((sin(y)*(e^x))+e^(-y))dx-((xe^-y)-((e^x)(cosy))dy=0 => Exact Equation which needs an integrating factor.

c. (x^2+y^2+x)dx+xydy=0 => Homogeneous equation

d. (y^2-xy)dx+x^2dy=0 which has an integrating factor of the form u(x,y)=(x^m)(y^n). => I am unsure what type of equation this is.

If anyone can help please advise. Post or reply to netapparition@yahoo.com
 
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  • #2
a. Bernoulli equation: The solution to the Bernoulli equation is given by:y(x) = c_1*exp(∫(y^2(cos(x)-sin(x))dx)+Cb. Exact Equation: The solution to the exact equation is given by:M(x,y) + N(x,y) = cWhere M and N are the derivatives of the left and right hand side of the equation respectively. To find M and N, multiply the entire expression by an integrating factor u(x,y):u(x,y)*(((sin(y)*(e^x))+e^(-y))dx-((xe^-y)-((e^x)(cosy))dy) = 0Then integrate both sides to find M and N:u(x,y)*((sin(y)*(e^x))+e^(-y)) - (x*e^-y)+((e^x)*(cosy)) = cc. Homogeneous equation:The solution to the homogeneous equation is given by:x^2+y^2+x = cd. I am not sure what type of equation this is.
 
  • #3


a. To solve this Bernoulli equation, first divide both sides by y^2:

(dy/dx)y^-2 + y^-1 = cos(x) - sin(x)

Then, let u = y^-1 and du/dx = -y^-2(dy/dx). Substituting this into the equation, we get:

-du/dx + u = cos(x) - sin(x)

This equation is now in the form of a linear differential equation, which can be solved by finding the integrating factor and using the method of integrating factors. The integrating factor is e^x, so multiply both sides by e^x:

e^x(-du/dx + u) = e^x(cos(x) - sin(x))

Then, use the product rule to simplify the left side:

d/dx(e^xu) = e^x(cos(x) - sin(x))

Integrating both sides with respect to x, we get:

e^xu = ∫e^x(cos(x) - sin(x))dx

Using integration by parts on the right side, we get:

e^xu = e^x(sin(x) + cos(x)) + C

Substituting back in u = y^-1, we get:

y^-1 = e^-x(sin(x) + cos(x)) + C

Solving for y, we get the general solution:

y = (e^-x(sin(x) + cos(x)) + C)^-1

b. To solve this exact equation, first check if the equation satisfies the exact equation condition:

Mx + Ny + ∂M/∂y = ∂N/∂x

Where M and N are the coefficients of dx and dy, respectively. In this case, M = (sin(y)e^x + e^-y) and N = -(xe^-y + e^xcos(y)). Taking partial derivatives, we get:

∂M/∂y = cos(y)e^x - e^-y

∂N/∂x = e^-y - e^xsin(y)

Since these are not equal, we need to find an integrating factor to make the equation exact. The integrating factor can be found by dividing the two partial derivatives and integrating with respect to either x or y. In this case, we will integrate with respect to x:

∫(∂M/∂y)/∂N
 
  • #4


Hello,

I am happy to assist with identifying and solving the types of differential equations mentioned in your post. After reviewing the equations provided, I can confirm that your identifications are correct. I have also included the notation and steps for solving each equation below.

a. (dy/dx)+y=y^2(cos(x)-sin(x)) => Bernoulli equation

To solve this equation, we can use the substitution u = y^(-1). This will transform the equation into a linear form, which can be easily solved using standard methods. The steps are as follows:

1. Substitute u = y^(-1) into the equation. This gives us: (dy/dx) + uy = cos(x) - sin(x)

2. Multiply both sides by u to get rid of the fraction. This gives us: (u(dy/dx)) + u^2y = u(cos(x) - sin(x))

3. Use the product rule to expand the first term on the left side. This gives us: (dy/dx) + u(dy/dx) + u^2y = u(cos(x) - sin(x))

4. Group the terms with dy/dx together and factor out u. This gives us: (dy/dx)(1 + u) + u^2y = u(cos(x) - sin(x))

5. Use the substitution u = y^(-1) to replace the u^2y term. This gives us: (dy/dx)(1 + u) + u(cos(x) - sin(x)) = u(cos(x) - sin(x))

6. Subtract u(cos(x) - sin(x)) from both sides to isolate the dy/dx term. This gives us: (dy/dx)(1 + u) = 0

7. Divide both sides by (1 + u) to solve for dy/dx. This gives us: dy/dx = 0

8. Integrate both sides with respect to x. This gives us: y = C, where C is a constant.

b. (((sin(y)*(e^x))+e^(-y))dx-((xe^-y)-((e^x)(cosy))dy=0 => Exact Equation which needs an integrating factor.

To solve this equation, we first need to check if it is exact. This can be done by checking if the partial derivatives of the terms with respect to x and y are
 

1. What is DE Identifying and how is it solved?

DE Identifying, or differential equation identifying, refers to the process of determining the type and order of a differential equation. It is solved by examining the variables and coefficients in the equation and using techniques such as integration, substitution, and separation of variables to simplify and solve the equation.

2. How is Homo solved in a scientific context?

In science, Homo refers to the genus of primates that includes modern humans. The process of solving Homo involves studying evolutionary history, genetics, and physical characteristics to understand the origins and development of the human species.

3. What is the significance of Bernoulli's equation in physics and how is it solved?

Bernoulli's equation is a fundamental equation in fluid mechanics that relates the pressure, velocity, and height of a fluid in a closed system. It is solved by using principles of conservation of energy and mass, as well as mathematical techniques such as integration and differentiation.

4. What is the difference between exact and inexact differential equations?

An exact differential equation is one in which the solution can be found by simple manipulation and integration, while an inexact differential equation requires additional steps such as integrating factors to solve. Exact differential equations also have the property that their partial derivatives are equal, while inexact equations do not.

5. How do scientists use differential equations in real-world applications?

Differential equations are used in a variety of fields in science, from physics and engineering to biology and economics. They are used to model and predict the behavior of physical systems, such as fluid flow and population growth, and to solve complex problems in various scientific disciplines.

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