# Recent questions in Differential equations

### Solve differential equation $$\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={\left({x}+{y}+{1}\right)}^{{2}}-{\left({x}+{y}-{1}\right)}^{{2}}$$

Laplace transform

### Use the appropriate algebra and Table of Laplace's Transform to find the given inverse Laplace transform. $$L^{-1}\left\{\frac{1}{(s-1)^2}-\frac{120}{(s+3)^6}\right\}$$

Laplace transform

### Determine the Laplace transform of the given function f. $$f(t)=(t -1)^2 u_2(t)$$

Laplace transform

### Solve differential equation $$\frac{\cos^2y}{4x+2}dy= \frac{(\cos y+\sin y)^2}{\sqrt{x^2+x+3}}dx$$

Laplace transform

### Find the laplace transform by definition. a) $$\displaystyle{L}{\left\lbrace{2}\right\rbrace}$$ b) $$\displaystyle{L}{\left\lbrace{e}^{{{2}{t}}}\right\rbrace}$$ c) $$\displaystyle{L}{\left[{e}^{{-{3}{t}}}\right]}$$

Laplace transform

### Find the solution of the initial value problem given below by Laplace transform $$y'-y=t e^t \sin t$$ $$y(0)=0$$

Differential equations

### Solve differential equation: $$\displaystyle{y}'+{y}^{{2}}{\sin{{x}}}={0}$$

Laplace transform

### Find the laplace transform of the following: $$a) t^2 \sin kt$$ $$b) t\sin kt$$

Laplace transform

### Solve differential equation $$dy/dx -12x^3y = x^3$$

Laplace transform

### use properties of the Laplace transform and the table of Laplace transforms to determine L[f] $$f(t)=e^{3t}\cos5t-e^{-t}\sin2t$$

Laplace transform

### use the Laplace transform to solve the given initial-value problem. $$y"+y=f(t)$$ $$y(0)=0 , y'(0)=1$$ where $$\displaystyle f{{\left({t}\right)}}={\left\lbrace\begin{matrix}{1}&{0}\le{t}<\frac{\pi}{{2}}\\{0}&{t}\ge\frac{\pi}{{2}}\end{matrix}\right.}$$

Laplace transform

### Find the Laplace transformation (evaluating the improper integral that defines this transformation) of the real valued function f(t) of the real variable t>0. (Assume the parameter s appearing in the Laplace transformation, as a real variable). $$f{{\left({t}\right)}}={2}{t}^{2}-{4} \cosh{{\left({3}{t}\right)}}+{e}^{{{t}^{2}}}$$

Laplace transform

### use properties of the Laplace transform and the table of Laplace transforms to determine L[f] $$f(t)=\int_0^t (t-w)\cos(2w)dw$$

Laplace transform