Separable differential equations and families of solutions Quiz

1. Find the general solution of $\frac{dy}{dx}=3y$.

• A. $y=3x+A$
• B. $y=Ae^{3x}$
• C. $y=e^{3x}+A$
• D. $y=\frac{A}{3x}$

2. Given $\frac{dy}{dx}=\frac xy$ and $y=2$ when $x=0$ (with $y>0$), find $y$.

• A. $y=x+2$
• B. $y=x^2+2$
• C. $y^2=x^2+2$
• D. $y^2=x^2+4$

3. The rate of decrease of the mass $m$ of a radioactive sample is proportional to $m$ (constant $k>0$). Which equation models this?

• A. $\frac{dm}{dt}=-km$
• B. $\frac{dm}{dt}=km$
• C. $\frac{dm}{dt}=-\frac km$
• D. $\frac{dm}{dt}=-kt$

4. What family of curves has differential equation $\frac{dy}{dx}=-\frac xy$?

• A. parabolas
• B. straight lines through the origin
• C. circles centred at the origin
• D. hyperbolas

5. Given $\frac{dy}{dx}=xe^{-y}$ and $y=0$ at $x=0$, find $y$ when $x=\sqrt2$.

• A. $\ln3$
• B. $\ln2$
• C. $\sqrt2$
• D. $\ln4$