Solving First Order Differential Equation using substitution

sristi89

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

defunc

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).

elibj123

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.

matematikawan

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.