# Differential equations questions and answers

Recent questions in Differential equations
First order differential equations

### Solve differential equation $$xydy−y^2dx= (x+y)^2 e^(−y/x)$$

Laplace transform

### Find Laplace transform of the given function $$te^{-4t}\sin 3t$$

Laplace transform

### Explain First Shift Theorem & its properties?

Laplace transform

### Find the inverse laplace transform of the function $$Y(s)=\frac{e^{-s}}{s(2s-1)}$$

First order differential equations

### Solve differential equation $$dy/dx= 15x^2 e^{-y}$$

Laplace transform

### Find the Laplace transform $$L\left\{u_3(t)(t^2-5t+6)\right\}$$ $$a) F(s)=e^{-3s}\left(\frac{2}{s^4}-\frac{5}{s^3}+\frac{6}{s^2}\right)$$ $$b) F(s)=e^{-3s}\left(\frac{2}{s^3}-\frac{5}{s^2}+\frac{6}{s}\right)$$ $$c) F(s)=e^{-3s}\frac{2+s}{s^4}$$ $$d) F(s)=e^{-3s}\frac{2+s}{s^3}$$ $$e) F(s)=e^{-3s}\frac{2-11s+30s^2}{s^3}$$

Laplace transform

### $$y''+y=e^{-2t}\sin t , y(0)=y′(0)=0$$ solution of given initial value problem with Laplace transform

Laplace transform

### Use the Laplace transform to solve the given integral equation $$(s)=2t^2+\int_0^t\sin\left[2(t-\tau)\right]x(\tau)d \tau$$

Second order linear equations

### Solve the Differential equations $$2y′′ + 3y′ − 2y = 14x^{2} − 4x − 11, y(0) = 0, y′(0) = 0$$

First order differential equations

### Solve differential equation $$y'+2y=y^2$$

Laplace transform

### Find the inverse Laplace transform $$f{{\left({t}\right)}}={L}^{ -{{1}}}{\left\lbrace{F}{\left({s}\right)}\right\rbrace}$$ of each of the following functions. $${\left({i}\right)}{F}{\left({s}\right)}=\frac{{{2}{s}+{1}}}{{{s}^{2}-{2}{s}+{1}}}$$ Hint – Use Partial Fraction Decomposition and the Table of Laplace Transforms. $${\left({i}{i}\right)}{F}{\left({s}\right)}=\frac{{{3}{s}+{2}}}{{{s}^{2}-{3}{s}+{2}}}$$ Hint – Use Partial Fraction Decomposition and the Table of Laplace Transforms. $${\left({i}{i}{i}\right)}{F}{\left({s}\right)}=\frac{{{3}{s}^{2}+{4}}}{{{\left({s}^{2}+{1}\right)}{\left({s}-{1}\right)}}}$$ Hint – Use Partial Fraction Decomposition and the Table of Laplace Transforms.

Laplace transform

### $$y_1'=-3y_1+y_2+u(t-1)e^t, y_2'=-4y_1+2y_2+u(t−1)e^t, y_1(0)=0, y_2(0)=21$$ Enclose arguments of functions, numerators, and denominators in parentheses.

Laplace transform

### Solve $$f(t)=e^t \cos t$$

Laplace transform

### Solve the following IVP using Laplace Transform $$y'-2y =1-t , y(0)=4$$

Laplace transform

### Calculate the Laplace transform $$L\left\{\sin(t-k) \cdot H(t-k)\right\}$$

Laplace transform

### Solve the Laplace transforms $$\dot x-2\ddot x+x=e^t t$$ $$\text{ given } t=0 \text{ and } x=0 \text{ and } x=1$$

Laplace transform

### Use integration by parts to find the Laplace transform of the given function $$f(t)=4t\cos h(at)$$

Laplace transform

### Use the Laplace transform to solve the given initial-value problem $${y}{''}+{2}{y}'+{y}={0},{y}{\left({0}\right)}={1},{y}'{\left({0}\right)}={1}$$

Laplace transform