Separable differential equations Quiz

1. Find the general solution of $\frac{\mathrm{d}y}{\mathrm{d}x}=3y$ for $y>0$.

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

2. Find the general solution of $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{x}{y}$.

• A. $y=\frac{x^2}{2}+c$
• B. $y=\ln x+c$
• C. $y=x+c$
• D. $y^2=x^2+A$

3. Which line separates the variables in $\frac{\mathrm{d}y}{\mathrm{d}x}=3x-3xy$?

• A. $\int\frac{1}{1-y}\,\mathrm{d}y=\int3x\,\mathrm{d}x$
• B. $\int\frac{1}{y}\,\mathrm{d}y=\int3x-3x\,\mathrm{d}x$
• C. $\int\frac{1}{3x}\,\mathrm{d}y=\int(1-y)\,\mathrm{d}x$
• D. $\int(1-y)\,\mathrm{d}y=\int3x\,\mathrm{d}x$

4. A population satisfies $\frac{\mathrm{d}P}{\mathrm{d}t}=0.2P$ with $P=50$ at $t=0$. Find $P$ when $t=5$.

• A. $100$
• B. $50.0$
• C. $136$
• D. $250$

5. For $y=Ae^{x^2}$ with $A>0$, which statement is true of every solution curve?

• A. It passes through the origin.
• B. It is symmetric about the $y$-axis and crosses it at $(0,A)$.
• C. It crosses the $x$-axis.
• D. It has a maximum at $x=0$.