# Recent questions in Differential equations

Differential equations

### The coefficient matrix for a system of linear differential equations of the form $$\displaystyle{y}^{{{1}}}={A}_{{{y}}}$$ has the given eigenvalues and eigenspace bases. Find the general solution for the system. $$\left[\lambda_{1}=-1\Rightarrow\left\{\begin{bmatrix}1 0 3 \end{bmatrix}\right\},\lambda_{2}=3i\Rightarrow\left\{\begin{bmatrix}2-i 1+i 7i \end{bmatrix}\right\},\lambda_3=-3i\Rightarrow\left\{\begin{bmatrix}2+i 1-i -7i \end{bmatrix}\right\}\right]$$

Differential equations

### Determine whether three series converges or diverges. if it converges, find its sum $$\displaystyle{\sum_{{{n}={0}}}^{{\infty}}}\frac{{{\left(-{1}\right)}^{{n}}}}{{4}^{{n}}}$$

Differential equations

### Solve the given system of differential equations. \[Dx+Dy+(D+1)z=0\) Dx+y=e^{t}\) Dx+y-2z=50\sin(2t)\)

Laplace transform

### Find the Laplace transforms of the given functions. $$\displaystyle g{{\left({t}\right)}}={4} \cos{{\left({4}{t}\right)}}-{9} \sin{{\left({4}{t}\right)}}+{2} \cos{{\left({10}{t}\right)}}$$

Laplace transform

### Use the Laplace transform to solve the heat equation $$u_t=u_{xx} 00$$ $${u}{\left({x},{0}\right)}= \sin{{\left(\pi{x}\right)}}$$ $${u}{\left({0},{t}\right)}={u}{\left({1},{t}\right)}={0} Laplace transform asked 2021-02-21 ### Solve the initial value problem below using the method of Laplace transforms. \(y"-16y=32t-8e^{-4t}$$ $$y(0)=0$$ $$y'(0)=15$$

Laplace transform

### Find the Laplace transforms of the following time functions. Solve problem 1(a) and 1 (b) using the Laplace transform definition i.e. integration. For problem 1(c) and 1(d) you can use the Laplace Transform Tables. a)$$f(t)=1+2t$$ b)$$f(t) =\sin \omega t \text{Hint: Use Euler’s relationship, } \sin\omega t = \frac{e^(j\omega t)-e^(-j\omega t)}{2j}$$ c)$$f(t)=\sin(2t)+2\cos(2t)+e^{-t}\sin(2t)$$

Differential equations

### Linear equations of second order with constant coefficients. Find all solutions on $$\displaystyle{\left(-\infty,+\infty\right)}.{y}\text{}-{2}{y}'+{5}{y}={0}$$

Laplace transform

### Find the inverse Laplace transform of (any two) i) $$\frac{(s^2+3)}{s(s^2+9)}$$ ii) $$\log\left(\frac{(s+1)}{(s-1)}\right)$$

Differential equations

### Solve differential equation $$dy/dx= (y^2-1)/(x^2-1)$$

Differential equations

### Find the differential of the function. $$\displaystyle{T}=\frac{{v}}{{1}}+{u}{v}{w}$$

Laplace transform

### Find the inverse Laplace transform of $$\frac{e^{-s}}{s(s+1)}$$

Laplace transform

### Find the inverse laplace trans. $$\displaystyle{F}{\left({s}\right)}=\frac{10}{{{s}{\left({s}^{2}+{9}\right)}}}$$

Laplace transform

### Use Laplace transform to solve the following initial-value problem $$y"+2y'+y=0$$ $$y(0)=1, y'(0)=1$$ a) \displaystyle{e}^{{-{t}}}+{t}{e}^{{-{t}}}\) b) \displaystyle{e}^{t}+{2}{t}{e}^{t}\) c) \displaystyle{e}^{{-{t}}}+{2}{t}{e}^{t}\) d) \displaystyle{e}^{{-{t}}}+{2}{t}{e}^{{-{t}}}\) e) \displaystyle{2}{e}^{{-{t}}}+{2}{t}{e}^{{-{t}}}\) f) Non of the above

Laplace transform

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