Ordinary Differential Equation

[stunner5000pt]

Homework Statement

Homework Statement

Solve $$\frac{dz}{dt} + 3 t e^{t+z} = 0$$

Homework Equations

None that I can think of...

The Attempt at a Solution

"Rearranging" the given question, we get:

$$\int \frac{dz}{e^z} = -3\int t e^t dt$$

$$-e^{-z} = -3 \left( t e^t - e^t \right) + C$$
$$e^{-z} = 3 \left( t e^t - e^t \right) + C$$
$$z = - ln \left( 3 t e^t - 3 e^t + C \right)$$

Is this all correct? The system into which I need to enter this answer is saying I am wrong :(

 

[Ray Vickson]

Homework Statement

Homework Statement

Solve $$\frac{dz}{dt} + 3 t e^{t+z} = 0$$

Homework Equations

None that I can think of...

The Attempt at a Solution

"Rearranging" the given question, we get:

$$\int \frac{dz}{e^z} = -3\int t e^t dt$$

$$-e^{-z} = -3 \left( t e^t - e^t \right) + C$$
$$e^{-z} = 3 \left( t e^t - e^t \right) + C$$
$$z = - ln \left( 3 t e^t - 3 e^t + C \right)$$

Is this all correct? The system into which I need to enter this answer is saying I am wrong :(

Maybe it does not like the '-' sign; have you tried entering
$$\ln \left( \frac{1}{3 t e^t - 3 e^t + C}\right)?$$

 

[stunner5000pt]

Maybe it does not like the '-' sign; have you tried entering
$$\ln \left( \frac{1}{3 t e^t - 3 e^t + C}\right)?$$

Let's hope its that... I only get one more shot :(

 

[MostlyHarmless]

I think you have to take the natural log of each part individually. $$lne^{-z}=ln3te^t-ln3e^t+lnc$$ that would give $$z=-ln3te^t+ln3e^t+lnc$$ Which simplifies to $$z=ln({\frac{c3e^t}{3te^t})}$$

 

[Ray Vickson]

I think you have to take the natural log of each part individually. $$lne^{-z}=ln3te^t-ln3e^t+lnc$$ that would give $$z=-ln3te^t+ln3e^t+lnc$$ Which simplifies to $$z=ln({\frac{c3e^t}{3te^t})}$$

No, you cannot do that. The answer I gave in my previous post was the one that Maple gave as the solution to the DE. The OP's workings were perfectly correct, as was the answer he gave.

 

[MostlyHarmless]

No, you cannot do that. The answer I gave in my previous post was the one that Maple gave as the solution to the DE. The OP's workings were perfectly correct, as was the answer he gave.

Ah, apologies.