Solving First Order Differential Equation using substitution

In summary: Then we can substitute in the values for the different variables. So in summary, our teacher gave us a hint that we should try using a substitution which is a system of sines, cosines, and looks something similar to 5.1sin(600*t)*u(t). I tried substituting: x(t)= A sin (w1*t)+B cos (w2*t)+ c cos(w3*t)*u(t) By differentiated this I get: x'(t)=A*w(1)*cos(w1*t)-B*w2*sin(w2*t)-c
  • #1
sristi89
8
0
Hi,

Here is the equation:

x+x'=5.1sin(600*t)*u(t)

Our teacher gave us a hint that we should try using a substitution which is a system of sines, cosines, and looks something similar to 5.1sin(600*t)*u(t).

I tried substituting:

x(t)= A sin (w1*t)+B cos (w2*t)+ c cos(w3*t)*u(t)

By differentiated this I get:

x'(t)=A*w(1)*cos(w1*t)-B*w2*sin(w2*t)-c*w3*sin(w3*t)*u(t)+c cos(w3*t)*u'(t)

Putting everything together I have:

A sin (w1*t)+B cos (w2*t)+ c cos(w3*t)*u(t)+A*w(1)*cos(w1*t)-B*w2*sin(w2*t)-c*w3*sin(w3*t)*u(t)+c cos(w3*t)*u'(t) =5.1sin(600*t)*u(t)

Then I get 5 equations:

1) A sin (w1*t)-B*w2*sin(w2*t)=0
2) B cos (w2*t)+A*w(1)*cos(w1*t)=0
3)c cos(w3*t)*u(t)=0
4)-c*w3*sin(w3*t)*u(t)=0
5)-c*w3*sin(w3*t)*u(t)=5.1sin(600*t)*u(t)

By solving 5, I get w3=600 and c=-.0085. As there are less equations than unknowns, how do I find A, B, wi, and w2.

Also, the initial condition is x(0)=0.

Thanks
 
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  • #2
An easy way to solve this problem is to multiply by e^t both sides, and rewriting the left then as d/dt(xe^t).
 
  • #3
i don't understand why you use w1,w2 and w3. clearly there is only one possible frequency for the solution so w1=w2=w3=600.
 
  • #4
sristi89 said:
Hi,

Here is the equation:

x+x'=5.1sin(600*t)*u(t)


Thanks


I think we need to be clear first about the function u(t) here. What it is. It is any function or it is a specific function such as the unit step function.
 

1. What is a first order differential equation?

A first order differential equation is an equation that involves the first derivative of an unknown function. It can be written in the form dy/dx = f(x,y), where y is the unknown function and f(x,y) is a function of both x and y.

2. How do you solve a first order differential equation using substitution?

The first step is to identify the type of substitution needed. This can be done by looking at the form of the equation and finding a function that can be substituted in for y or x. Then, substitute the function into the equation and solve for the remaining variable. Finally, integrate both sides of the equation and solve for the constant of integration.

3. What are the benefits of using substitution to solve a first order differential equation?

Using substitution allows for a simpler and more systematic approach to solving differential equations. It can also help to reduce the complexity of the equation and make it easier to solve.

4. What are some common mistakes when solving first order differential equations using substitution?

One common mistake is not choosing the correct substitution, which can lead to a more complex equation. Another mistake is not properly substituting the function into the equation, which can result in incorrect solutions.

5. Can all first order differential equations be solved using substitution?

No, not all first order differential equations can be solved using substitution. Some equations may require more advanced techniques, such as separation of variables or the method of integrating factors.

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