anonymous
  • anonymous
how to solve dy/dt+2y=exp(t/2)
Mathematics
  • Stacey Warren - Expert brainly.com
Hey! We 've verified this expert answer for you, click below to unlock the details :)
SOLVED
At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.
schrodinger
  • schrodinger
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
bahrom7893
  • bahrom7893
Use an inegrating factor: This equation is in the format: \[dy/dt + 2y = e^{t/2}\] \[dy/dt + P(x)y = Q(x)\]
bahrom7893
  • bahrom7893
Use an inegrating factor: This equation is in the format: dy/dt + 2y = e^{t/2} dy/dt + P(t)y = Q(t) I.f. = e^(int[P(t)dt])
bahrom7893
  • bahrom7893
I.f. = e^[Integral(2dt)] = e^(2t) Multiply your diff equation by the integrating factor e^(2t) e^(2t) * [dy/dt+2y] = e^(2t) * e^(t/2)

Looking for something else?

Not the answer you are looking for? Search for more explanations.

More answers

bahrom7893
  • bahrom7893
I.f. = e^[Integral(2dt)] = e^(2t) Multiply your diff equation by the integrating factor e^(2t); e^(2t) * [dy/dt+2y] = e^(2t) * e^(t/2)
bahrom7893
  • bahrom7893
Sorry for double post my pc is having issues. Now simplify: (dy/dt)*e^(2t) + 2y*e^(2t) = e^(2t + [t/2]); Notice the left side is a product rule for derivatives: (d/dt)[y*e^2t] = e^(5t/2); {Btw if you don't understand how i got the left side; differentiate that product with respect to t, you just have to recognize the form and the reason why we use integrating factors is because we want to put one side of the d.e. into that form: y*2*e^(2t) + e^(2t)*(dy/dt)}
bahrom7893
  • bahrom7893
So now you know that: (d/dt)[y*e^(2t)]=e^(5t/2) {derivative of y*e^(2t) with respect to t is equal to e^(5t/2)} Integrate both sides with respect to t: (d/dt)[y*e^(2t)]=e^(5t/2) Int{ (d/dt)[y*e^(2t)] * dt}=Int{ e^(5t/2) * dt }
bahrom7893
  • bahrom7893
Integral of the derivative is just the original function (i think it is called newton's method or something like that). So the integral of the left side is: y*e^(2t) = Integral(e^[5t/2]*dt); the right side is easy to integrate: Integral(e^[5t/2]*dt); Let u = 5t/2; then du = (5/2) dt; Multiply the integral by (2/5)*(5/2) - Basically im multiplying by one, i just need that 5/2 for the du; so: Integral(e^[5t/2]*dt) = (2/5)* Integral(e^u * du) = (2/5)*e^u +C
bahrom7893
  • bahrom7893
We let u = 5t/2, so: y*e^(2t) = (2/5)*e^u +C y*e^(2t) = (2/5)*e^(5t/2) +C
bahrom7893
  • bahrom7893
Simplify (divide everything by e^(2t)): y*e^(2t) = (2/5)*e^(5t/2) + C y = (2/5) * e^([5t/2] - [2t]) + C * e^(-2t) y = (2/5) * e^(t/2) + C*e^(-2t) <= Your answer
bahrom7893
  • bahrom7893
Simplify (divide everything by e^(2t)): y*e^(2t) = (2/5)*e^(5t/2) + C; y = (2/5) * e^([5t/2] - [2t]) + C * e^(-2t); y = (2/5) * e^(t/2) + C*e^(-2t) <= Your answer
bahrom7893
  • bahrom7893
Sorry for double post again, im havin issues with pc.. So that's the method u use to solve these types of problems. My answer should be correct, just check my arithmetic, i had a class early in the morning and have a break now solved this without a calculator, but i think there shouldn't be any mistakes, but just in case check the work. But that's how u solve those problems. Btw, my next class is differential equations!

Looking for something else?

Not the answer you are looking for? Search for more explanations.