Find the Inverse Laplace Transforms

Find the inverse Laplace transform of: 3s+5s(s2+9) Workings: 3s+5s(s2+9) =3ss(s2+9+5s(s2+9) =3s2+9+5s1s2+9 =sin(3t)+5s1s2+9 Now I’m not to sure on what to do. Any help will be appreciated. Answer The convolution theorem is, found here, L−1{f(s)g(s)}=∫t0f(t−u)g(u)du In the case here f(s)=1/s which is the transform of 1 and g(s) being the transform of sin. Now L−1{1s(s2+a2)}=1a∫t0(1)sin(au)du=−1a[cos(au)a]t0=−1a2(cos(at)−1) With … Read more

Find all line equations that are tangent to x3−xx^3 – x and pass through (−2,2)(-2,2)

So I have the equation: f(x)=x3−x So we know that the slope of the curve for some x is given by: f′(x)=3×2−1 And need to find equations of lines that are tangent to that curve, and also pass through the point (−2,2). I have seen a similar question involving a circle equation, but couldn’t get … Read more

Why $x^2$ may not be a solution to a general second order homogenous differential equations

I’m relatively new to differential equations and I recently read that if the functions $p(x)$ and $q(x)$ are continuous at $x=0$ then it is never possible for $x^2$ to be a solution to the equation $y”+p(x)y’+q(x)y = 0$ Given that this is still relatively elementary work as I’ve only just started looking at second order … Read more

How to solve this ODE: $\ddot{y}+\frac{1}{2f(y)}\frac{df(y)}{dy}(\dot{y})^2=0$ [closed]

Closed. This question is off-topic. It is not currently accepting answers. Want to improve this question? Update the question so it’s on-topic for Mathematics Stack Exchange. Closed 6 years ago. Improve this question Initial conditions are $y(0)=A,\dot{y}(0)=B$, and $f(\cdot)>0$ is a given function. Thanks! Answer Multiply everything by $2f(y) \dot{y}$. Then you have $$ f(y) … Read more

Prove that the solution of y′+y=arctan(ex),y(0)=2y’+y=\arctan(e^x), y(0)=2 admits horizontal asymptote.

Let us consider the Cauchy problem: y′+y=arctan(ex),    y(0)=2 Prove that the function y(x) admits horizontal asymptote without solving the problem. Answer I’ll prove a more general case, instead. If limx→+∞f(x)+f′(x)=l, then: \begin{align} &\lim_{x\rightarrow+\infty}f(x)=l\\ &\lim_{x\rightarrow+\infty}f'(x)=0 \end{align} Suppose we have \lim_{x\rightarrow+\infty}f(x)+f'(x)=l Then we have e^x(f(x)+f'(x))\approx_{x\rightarrow +\infty} le^x (e^xf(x))’\approx_{x\rightarrow +\infty} le^x=(le^x)’ Since x\rightarrow le^x is positive and its integral … Read more

Solve the following second-order differential equation: $\ddot{x} + \dot{x} = 5t\cos(t) + 4\sin(t)$

I am trying to solve the following second-order differential equation: $$\ddot{x} + \dot{x} = 5t\cos(t) + 4\sin(t). (*)$$ I know that if the equation had instead been: $$\ddot{x} + \dot{x} = 5t[\cos(t) + 4\sin(t)],$$ then I could have tried a solution of the form: $$x(t) = P_1(t)\cos(t) + P_2(t)\sin(t),$$ where $P_1(t), P_2(t)$ were linear polynomials. … Read more

First-order linear ordinary differential equation with piecewise constant source term

Find a continuous solution satisfying the DE: dydx+2y=f(x) f(x)={1,0≤x≤30,x>3.y(0)=0 I don’t get this problem at all. Can anyone explain what the above means for starters? Answer solve the equation in two pieces and then match it at the boundary x=1. the two problems ares y′+2y=1,y(0)=0 has the solution y=12(1−e−2x),x≤3. now solve y′+2y=0,y(3)=12(1−e−6). the solution is … Read more