DIFFERENTIAL EQUATIONS (MMR w4)

What is the integral factor,

 $V\left(x\right)$  for the differential equation  $x\frac{dy}{dx}-2y=x^3\ln x$  ?

•  $e^{2x}$  

•  $e^{-2x}$  

•  $\frac{1}{x^2}$  

•  $x^2$  

From the list below, choose 2 first order linear differential equations.

• $\frac{dy}{dx}+2y=\sin x$

• $x^2+\left(y+2\right)_{ }^3\frac{dy}{dx}=0$

• $y\sin^2x\ \frac{dy}{dx}=\frac{1}{2}\cos x$

• $\frac{\text{d}\theta}{\text{d}t}-\frac{6\theta t}{3t^2+2}=\sqrt{t^3+2t}$

Which of the following is the CORRECT standard form of the differential equation $\cos x\ \frac{dy}{dx}+y\ \sin x=e^x\cos^2x$  ?

•  $\frac{dy}{dx}+y\ \cot\ x=\cos x$  

•  $\frac{dy}{dx}+y\ \frac{\sin x}{\cos x}=e^x\cos^3x$  

•  $\frac{\text{d}y}{\text{d}x}+y\tan x=e^x\cos x$  

•  $\frac{dy}{dx}-y\ \frac{\sin x}{\cos x}=e^x$  

Which of the following is the TRUE steps involved in solving the first order linear differential equation $x\ \frac{dy}{dx}+3y=e^x$  ?

•  $\int\frac{\text{d}}{\text{d}x}ye^{3x}\ dx=\int e^{4x}dx$  

•  $x^3\ \frac{dy}{dx}+3x^2y=x^2e^x$  

•  $V\left(x\right)=e^{3x}$  

•  $\frac{\text{d}y}{\text{d}x}+\frac{3}{y}=e^x$  

Which of the following is true to represent the integral factor multiply with standard form differential equation for

 $x\ \frac{dy}{dx}-2y=x^3\ \ln x\ ?$  

•  $\frac{dy}{dx}\frac{1}{x^2}-\frac{2}{x^3}y\ =\ln x$  
  

•  $\frac{1}{x^2}\ \frac{dy}{dx}-\frac{2y^2}{x^3}=\frac{\ln x}{x}$  
  

•  $\frac{1}{x}\frac{dy}{dx}-\frac{2y}{x^2}=x\ln x$  

Find the solution for

 $\cos x\ \frac{\text{d}y}{\text{d}x}+y\sin x=e^x\cos^2x$  given that  $x=0$  when  $y=2$  .

•  $y=\left(e^x+A\right)\cos x$  

•  $y=\cos x\left(1+e^x\right)$  

•  $y=-e^x\left(2\tan x+1\right)$  

Which of the implementation of product rule is CORRECT to the respective first order differential equation?

• DE: $\frac{ds}{dt}-\frac{3s}{t}=t^2$ ;
  PR: $\frac{d}{dt}s\ t^3=t^5$

• DE: $\frac{dy}{dx}+\frac{y}{2x}=\sqrt{x}$ ;
  PR: $\frac{d}{dx}y\sqrt{x}=x$

• DE: $\frac{du}{dv}-\frac{2u}{v}+1=0$ ;
  PR: $\frac{d}{dv}u\ v^2=-v^2$

If $y=\frac{1}{e^{2x}}\left[\frac{x^4}{4}+\ln\left|x\right|+c\right]$  is the general solution to a certain differential equation, find the value of  $c$ , given that  $x=1,\ \ y=0$  .

• 0

•  $-\frac{1}{4}$  

•  $\ln4$  

• any value in Real Number

What is the integral factor, $V\left(x\right)$  for the differential equation  $\frac{dy}{dx}+y\cot x=\operatorname{cosec}x\tan^3x$  ?

•  $\tan x$  

•  $-\operatorname{cosec}^2x$  

•  $\frac{1}{\sin x}$  

•  $\sin x$  

Find the general solution for $\frac{d\theta}{dt}-3\theta=2e^{3t}$  .

•  $\theta=\frac{2+c}{e^{3t}}$  
  

•  $\theta=e^{-3t}\left(2t+c\right)$  
  

•  $\theta=e^{3t}\left(2t+a\right)$