Solving Differential Response Problems in EE Circuit Analysis

[thundercleese]

I get close to the answer but just not seeing where the integers in the answers are coming from, perhaps forgetting a constant? This is a EE circuit analysis class.

Homework Statement

Given y(t) = dx(t)/dt
a. Find x(t) if y(t)=

0 : t<0;​

e^{-t} : t>0​

b. Find x(t) for t=1 if x(1)=1 and y(t)=t^{2} for t\geq1

The Attempt at a Solution

Part a
y(t)=\frac{dx(t)}{dt}

when y(t) = 0:
∫0 = ∫\frac{dx(t)}{dt}
0 = x(t) (this part agrees with given answer)

when y(t) = e^{-t}:
∫ e^{-t}=∫\frac{dx(t)}{dt}
-e^{-t}=x(t)

part b, y(t)=t^{2}:

∫t^{2} = ∫\frac{dx(t)}{dt}
\frac{1}{3}t^{3} = x(t)

Given answers:

part a: 0, t\leq0; (1-e^{-t}), t\geq0

part b: \frac{1}{3}(2+t^{3}), t \geq1

 

[berkeman]

I get close to the answer but just not seeing where the integers in the answers are coming from, perhaps forgetting a constant? This is a EE circuit analysis class.

Homework Statement

Given y(t) = dx(t)/dt
a. Find x(t) if y(t)=

0 : t<0;​

e^{-t} : t>0​

b. Find x(t) for t=1 if x(1)=1 and y(t)=t^{2} for t\geq1

The Attempt at a Solution

Part a
y(t)=\frac{dx(t)}{dt}

when y(t) = 0:
∫0 = ∫\frac{dx(t)}{dt}
0 = x(t) (this part agrees with given answer)

when y(t) = e^{-t}:
∫ e^{-t}=∫\frac{dx(t)}{dt}
-e^{-t}=x(t)

part b, y(t)=t^{2}:

∫t^{2} = ∫\frac{dx(t)}{dt}
\frac{1}{3}t^{3} = x(t)

Given answers:

part a: 0, t\leq0; (1-e^{-t}), t\geq0

part b: \frac{1}{3}(2+t^{3}), t \geq1

Welcome to the PF.

Yeah, you have to include the constant from the initial condition x(1)=1. Plug t=1 into your solution for b, and plug it into the given solution. Do you see the difference?

 

[thundercleese]

I do, my answer resolves to 1/3 whereas the correct answer resolves to 1. My calculus is a bit rusty but adding the +C for the constant of integration didn't get me any closer to the 1 in part a or the 2 in part b.

 

[berkeman]

I do, my answer resolves to 1/3 whereas the correct answer resolves to 1. My calculus is a bit rusty but adding the +C for the constant of integration didn't get me any closer to the 1 in part a or the 2 in part b.

Could you show your work on adding the constant in part b? The answer given matches the initial condition, and your answer should match it too. The form of their answer looks a little weird, but if you expand terms, it goes into the standard form.

 

[thundercleese]

I didn't get beyond adding a C to the right side. I think the problem may be with the limits of the integration but I'm not sure, hence my post here.