Differential Equations Review

Which of the following differential equations match the graph above?

 

•  $\frac{\text{d}y}{\text{d}x}=y^2-1$  

•  $\frac{\text{d}y}{\text{d}x}=x^2-1$  

•  $\frac{\text{d}y}{\text{d}x}=y^2+1$  

•  $\frac{\text{d}y}{\text{d}x}=x^2+1$  

Suppose y=f(x) is a particular solution to the differential equation dy/dx=x–y such that f(0)=0. Use the slope field above to estimate the value of f(2).

• -2

• -1

• 0

• 1

Match the following differential equation with the correct slope field: dy/dx=x–y^2.

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• 
• 

Shown above is a slope field for which of the following differential equations?

• $\frac{dy}{dx}=1+x$

• $\frac{dy}{dx}=x^2$

• $\frac{dy}{dx}=x+y$

• $\frac{dy}{dx}=\frac{x}{y}$

Which of the following is the solution to the differential equation dy/dx=(x^2)/y with initial condition y(3)=-2?

• $y=2e^{\frac{-9+x^3}{3}}$

• $y=-2e^{\frac{-9+x^3}{3}}$

• $y=\sqrt{\frac{2x^3}{3}}$

• $y=\sqrt{\frac{2x^3}{3}-14}$

• $y=-\sqrt{\frac{2x^3}{3}-14}$

Solve the differential equation dy/dx=e^2x in terms of y given the condition that y(0)=0.

• $y=\frac{1}{2}e^{2x}+\frac{1}{2}$

• $y=\frac{1}{2}e^{2x}-\frac{1}{2}$

• $y=\frac{1}{2}e^{2x}+\frac{1}{3}$

• $y=\frac{1}{2}e^{2x}-\frac{1}{3}$

• $y=\frac{1}{2}e^x-\frac{1}{2}$

Solve the differential equation dy/dx=y/1+x^2 in terms of y given the condition that y(0)=1.

• $y=\ln\left|1+x^2\right|$

• $y=2\ln\left|1+x^2\right|$

• $y=e^{2\arctan\left(x\right)}$

• $y=e^{\arctan\left(x\right)}$

• $y=e^{\arctan\left(2x\right)}$

Which of the following is the solution to the differential equation dy/dx=-x/ye^x^2 with initial condition (0,1)? Assume y>0.

• $e^{-x^2}$

• $e^{x^2}$

• $e^{\frac{-x^2}{2}}$

• $e^{\frac{x^2}{2}}$

• $-e^{\frac{x^2}{2}}$

Find the general solution of the differential equation y’–ysin(x)=0.

• $|y|=Ce^{-\sin\left(x\right)}$

• $|y|=Ce^{\sin\left(x\right)}$

• $|y|=Ce^{-\cos\left(x\right)}$

• $|y|=Ce^{\cos\left(x\right)}$

• $|y|=-Ce^{\cos\left(x\right)}$

Find the general solution of the differential equation dy/dx=2x/4+x^2.

• $y=\arctan(2x)+C$

• $y=\frac{1}{2}\arctan(2x)+C$

• $y=\arctan(x^2+4)+C$

• $y=\ln|x^2+4|+C$

• $y=2x\ln|x^2+4|+C$