Solving an ODE: Seeking c so y'(0)=0

• MHB
C= (1/0.28)B and y= (1/0.28)Be^{-5.6x}+ Acos(2x)+ Bsin(2x).In summary, the conversation is about solving the equation y'+5.6y=9.5cos(2x)+2.4sin(2x) and finding the constant (c) in order to satisfy the condition y'(0)=0. The method of undetermined coefficients is used to find the solution, which is y= (1/0.28)Be^{-5.6x}+ Acos(2x)+ Bsin(2x).
so I am trying to solve this equation y'+5.6y=9.5cos(2x)+2.4sin(2x) . I want the c in order to y'(0)=0. I am really lost

Do you know how to do the method of undetermined coefficients? Please let us know what you've done.

-Dan

so I am trying to solve this equation y'+5.6y=9.5cos(2x)+2.4sin(2x) . I want the c in order to y'(0)=0. I am really lost
There is NO "c" in what you wrote! Do you mean the constant in the solution? It isn't necessarily called "c"!

The general solution to the associated homogenous equation, y'+5.6y= 0, is $$\displaystyle y= Ce^{-5.6x}$$. To find a solution to the entire equation, let $$\displaystyle y= Acos(2x)+ Bsin(2x)$$. Then $$\displaystyle y'= -2Asin(2x)+ 2Bcos(2X)$$ and the $$\displaystyle y'+ 5.6y= -2Asin(2x)+ 2Bcos(2x)+ 5.6Acos(2x)+5.6Bsin(2x)= (-2A+ 5.6B)sin(2x)+ (2B+ 5.6A)cos(2x)= 2.4sin(2x)+ 9.5cos(2x)$$.

Since this to be true for all x, we must hav -2A+ 5.6B= 2.4 and 2B+ 5.6A= 9.5.

Solve those two equations for A and B. Then the solution to the entire equation is $$\displaystyle y= Ce^{-5.6x}+ Acos(2x)+ Bsin(2x)$$ for those A and B. Then $$\displaystyle y'= -0.56Ce^{-5.6x}- 2Asin(2x)+ 2Bcos(2x)$$ so that $$\displaystyle y'(0)= -0.56C+ 2B= 0$$. $$\displaystyle C= (2/0.56)B$$ where B was found before.

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1. What is an ODE?

An ODE, or ordinary differential equation, is a mathematical equation that describes the relationship between a function and its derivatives. It involves one or more independent variables and the derivatives of the dependent variable with respect to those independent variables.

2. What does it mean to solve an ODE?

Solving an ODE means finding a function that satisfies the given differential equation. This involves finding the value of the unknown function at different points in the domain, usually through the use of mathematical techniques and algorithms.

3. Why is it important to find c in an ODE to satisfy y'(0)=0?

The constant c in an ODE represents the initial condition of the function at a specific point in the domain. In this case, y'(0)=0 means that the derivative of the function at the point 0 is equal to 0. Finding the correct value of c ensures that the solution satisfies this specific initial condition.

4. What are some methods for solving an ODE and finding c?

There are several methods for solving an ODE, including separation of variables, substitution, and using integrating factors. The method used to find c will depend on the specific form of the ODE and its initial conditions.

5. Can an ODE have multiple solutions for c?

Yes, an ODE can have multiple solutions for c. This is because the value of c affects the overall solution of the ODE and can result in different functions that satisfy the given differential equation. It is important to carefully consider the initial conditions and choose the appropriate value of c for the desired solution.

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