Separation of Variables [AP CALC AB] 7.6/7.7

What is the general solution to the differential equation  $\frac{dy}{dx}=\frac{\left(x-1\right)}{3y^2}$  for  $y>0$  ?

•  $y=\frac{1}{-\frac{3}{2}x^2+3x+C}$  

•  $y=\left(e^{\left(\frac{3x^2}{2}-3x+C\right)}\right)^{\frac{1}{2}}$  

•  $y=\left(x^2-x\right)^{\frac{1}{3}}+C$  

•  $y=\left(\frac{x^2}{2}-x+C\right)^{\frac{1}{3}}$  

What are all solutions to the differential equation $\frac{dy}{dx}=\sec^2x$  ?

•  $y=\tan\ x\ +C$  

•  $y=\sec\ x\ +\ C$  

•  $y=\frac{1}{3}\sec^3x+C$  

•  $y=2\sec^2x\tan\ x\ +C$  

Tags

CCSS.HSA-REI.B.4B

What is the general solution to the differential equation  $\frac{dy}{dx}=\frac{x\cos\left(x^2\right)}{4y}$  for  $y>0$   ?

•  $y=\frac{1}{2}\sqrt{\sin\left(x^2\right)}+C$  

•  $y=\sqrt{\frac{1}{4}\sin\left(x^2\right)+C}$  

•  $y=\frac{1}{8}\sin\left(x^2\right)+C$  

•  $y=Ce^{\frac{1}{8}\sin\left(x^2\right)}$  

Which of the following is the solution to the differential equation  $\frac{dy}{dx}=\left(x-2\right)\left(y-2\right)$   for  $y>2$   with the initial condition  $y\left(4\right)=5$   ?

•  $y=4+e^{\left(\frac{x^2}{2}-2x\right)\ }$  

•  $y=2+3e^{\left(\frac{x^2}{2}-2x\right)}$  

•  $y=\frac{4+\sqrt{16+4\left(x^2-4x+5\right)}}{2\ }\$  

•  $y=\frac{-x^2+4x+5}{1-\frac{x^2}{2}+2x}$  

Which of the following is the solution to the differential equation  $\frac{dy}{dx}=y^2-\left(xy\right)^2\$   with the initial condition  $y\left(3\right)=1$   ?

•  $y=\frac{5}{6}-\frac{3}{3x-x^3}$  

•  $y=-\frac{3}{3x-x^3+15}$  

•  $y=\left(3x-x^3+19\right)^{\frac{1}{3}}$  

•  $y=\left(e^{\left(x-\frac{1}{3}x^3+6\right)}\right)^{\frac{1}{2\ }}$  

Let  $y=f\left(x\right)$  be the particular solution to the differential equation  $\frac{dy}{dx}=y^2$  with the initial condition  $f\left(1\right)=1$  .  Which of the following gives an expression for 

•  $f\left(x\right)=\frac{\left(2x-1\right)}{x}\ for\ x\ne0$  

•  $f\left(x\right)=\frac{\left(2x-1\right)}{x}\ for\ x>0$  

•  $f\left(x\right)=\frac{1}{2-x}\ \ for\ \ x\ne2$  

•  $f\left(x\right)=\frac{1}{\left(2-x\right)}\ for\ \ x<2$  

A student attempts to solve the DEQ  $\frac{dy}{dx}=x^3\$   with the initial condition of   $\left(0,2\right)$   Find the error.

• Step 1

• Step 2

• Step 3

• Step 4

Which of the following is the solution to the differential equation  $\frac{dy}{dx}=\frac{2y}{2x+1}$   with the initial condition  $y\left(0\right)=e$  for  $x>-\frac{1}{2}$   ?

•  $y=e^{\left(2x^2+2x+1\right)}$  

•  $y=2ex+e$  

•  $y=\sqrt{x^2+x+e^2}$  

•  $y=\sqrt{\frac{1}{2}\ln\left(2x+1\right)+e^2}$  

If  $\frac{dy}{dx}=y\sec^2x$   and  $y=5$   when  $x=0$  , then  $y=$  

•  $e^{\tan x}+4$  

•  $e^{\tan\ x}+5$  

•  $5e^{\tan x}$  

•  $\tan x+5$  

•  $\tan\ x+5e^{x\ }$  

Which of the following is the solution to the differential equation  $\frac{dy}{dx}=\frac{4x}{y}$  , where  $y\left(2\right)=-2$   ?

•  $y=2x$   for  $x>0$  

•  $y=2x-6$   for  $x\ne3$  

•  $y=-\sqrt{4x^2-12}$   for  $x>\sqrt{3}$  

•  $y=\sqrt{4x^2-12}$   for  $x>\sqrt{3}$  

•  $y=-\sqrt{4x^2-6}$   for   $x>\sqrt{1.5}$