Recent content by 1s1

This integration rule seems really great, but I can't seem to find it in my undergraduate advanced engineering mathematics textbook. Either I don't know what I'm looking for or I need to upgrade textbooks :)

Thanks for the help! $$\mathcal{L}^{-1}\Big\{\frac{s}{s^2+1}\Big\} = cos(t)$$ $$\mathcal{L}^{-1}\Big\{\frac{1}{s^2+1}\Big\} = sin(t)$$ ##f(t)=cos(t)## and ##g(t)=sin(t)## So you can use ##\mathcal{L}^{-1}\Big\{F(s)G(s)\Big\} = f*g## and do the convolution ##cos(t)*sin(t)##...

Homework Statement ##\mathcal{L}^{-1}\Big\{\frac{s}{(s^2+1)^2}\Big\}## I'm trying to figure out how to find the inverse Laplace transform of this expression. Is this something you just look up in a table or is there a way to find it directly, maybe by Convolution?

This is very helpful, thanks!

You may be right about the relevant equation being negative on the RHS. My book has it written as I did, but my prof mentioned that physicists sometimes make sign changes to the constant. But I am still confused about how to make the function into a function of position? I assume you use...

Homework Statement An object of mass ##5##kg is released from rest ##1000##m above the ground and allowed to fall under the influence of gravity. Assuming the force due to air resistance is proportional to the velocity of the object with proportionality constant ##b=50##N-sec/m, determine the...

Making the suggested substitution: ##u=x+y+3## and using: $$\int \frac{1}{a^{2}+u^{2}}du=\frac{1}{a}tan^{-1}\frac{u}{a}+C$$ $$x=tan^{-1}(x+y+3)+C$$ The choice of substitution in this case seems similar to what you would choose when doing integration by substitution, so hopefully that will...

Thanks! Clearly, I don't understand how substitution works ... yet. The only substitution we've learned so far is for Bernoulli's Equation, ##\frac{dy}{dx}+P(x)y=f(x)y^{n}## Where the substitution is ##u=y^{1-n}## I'll run ##u=x+y+3## through and I should get a linear equation like you...

Homework Statement Find the general solution: $$\frac{dy}{dx}=(x+y+3)^{2}$$ Homework Equations The Attempt at a Solution Methods I have learned: separation of variables, integrating factor for linear equations, exact equations, and substitution. I don't even know where to begin...

Ha, ha good point! I suppose simple separation of variables would be SO much simpler. By the way, I'm new to working with ODE's, is using integrating factors pretty typical or are there alternative methods that are more common? Guess I'll probably find out more as I learn more about diffEq.

Figured it out ... thanks Zondrina! Have to appeal to the definition of continuity at a point. So, you can say that ##Ce^{-1}=1## $$C=e$$ And the solution for ##x \geq 0## $$y=\frac{e}{e^{x}}$$ Thanks!

Good point! I suppose I would need a different initial condition for this function?

Homework Statement $$\frac{dy}{dx}+y=\left\{\begin{matrix}1, \ 0\leq x< 1 \\ 0, \ x\ge1 \ \ \ \ \ \ \ \end{matrix}\right.$$ Homework Equations The Attempt at a Solution $$P(x)=1$$ Integrating factor ##=e^{x}## For ##f(x)=1##: $$\frac{d}{dx}[e^{x}y]=e^{x}$$ Integrating...

Thanks! When I divided through by ##sin(x)## from the step you suggested, I had forgotten to divide ##C## by ##sin(x)##. I tend to ignore the constants which is a big mistake. Thanks for the help!

Homework Statement Solve the initial value problem: $$sin(x)y' + ycos(x) = xsin(x), y(2)= \pi/2$$ Homework Equations The Attempt at a Solution Recognizing it as a Linear First-Order Equation:$$\frac{dy}{dx}+y\frac{cosx}{sinx}=x$$ $$P(x)=\frac{cosx}{sinx}$$ Integrating...