# Recent questions in Differential equations

Differential equations

### Solve differential equation $$\displaystyle{y}'+{y}{\cot{{\left({x}\right)}}}={\sin{{\left({2}{x}\right)}}}$$

Laplace transform

### Solve by Laplace transforms $$y"-3y'+2y=2\delta(t-1)$$ $$y(0)=1$$ $$y'(0)=0$$

Laplace transform

### Solve differential equation $$(2y-e^x)dx+xdy=0$$, x>0

Differential equations

### Let $$y_1$$ and $$y_2$$ be solution of a second order homogeneous linear differential equation $$y''+p(x)y'+q(x)=0$$, in R. Suppose that $$y_1(x)+y_2(x)=e^{-x}$$, $$W[y_1(x),y_2(x)]=e^x$$, where $$W[y_1,y_2]$$ is the Wroian of $$y_1$$ and $$y_2$$. Find p(x), q(x) and the general form of $$y_1$$ and $$y_2$$.

Laplace transform

### Find x(t) and Y(t) using Laplace transform. $$\begin{cases} \frac{dx}{dt}=2x-3y \frac{dy}{dt}=y-2x \end{cases}$$ $$x(0)=8 , y(0)=3$$

Laplace transform

### determine the inverse Laplace transform of the function. $$\displaystyle{Q}{\left({s}\right)}=\frac{s}{{{s}^{2}+{64}}}$$

Laplace transform

### Use Laplace transform to evaluate the integral $$\int_0^\infty te^{-2t} \sin(2t)dt$$ a) $$\frac{1}{B}$$ b) not defined c) $$\frac{4s}{(s^2+4)^2}$$ d) $$\frac{4(s+2)}{(s^2+4s+8)^2}$$

Laplace transform

### Solve the third-order initial value problem below using the method of Laplace transforms $$y'''-2y"-21y'-18y=-18$$ $$y(0)=2$$ $$y'(0)=7$$ $$y"(0)=95$$

Laplace transform

### Solve differential equation $$xy'= 6y+12x^4 y^(2/3)$$

Differential equations

### A particle moves along the curve $$\displaystyle{x}={2}{t}^{{2}}{y}={t}^{{2}}-{4}{t}$$ and z=3t-5 where t is the time.find the components of the velocity at t=1 in the direction i-3j+2k

Laplace transform

### How to solve this equation $$y'''-4y"+2y'-16y=4x+1$$ using Method of Undetermined Coefficient, Variation of Parameters and Laplace Transformation

Laplace transform

### Given that $$f{{\left({t}\right)}}={4}{e}^{{-{3}{\left({t}-{4}\right)}}}$$ a) Find $${L}{\left[\frac{{{d} f{{\left({t}\right)}}}}{{{\left.{d}{t}\right.}}}\right]}$$ by differentiating f(t) and then using the Laplace transform tables in lecture notes. b) Find $${L}{\left[\frac{{{d} f{{\left({t}\right)}}}}{{{\left.{d}{t}\right.}}}\right]}$$ using the theorem for differentiation c) Repeat a) and b) for the case that $$f{{\left({t}\right)}}={4}{e}^{{-{3}{\left({t}-{4}\right)}}}{u}{\left({t}-{4}\right)}$$

Laplace transform