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A student finds the general solution to \(y”+ y=0\) and gives \(y=e^t\). What is the issue?

We want to solve \(y’+\frac{{2}}{{x}}y=\frac{{1}}{{x^2}}\cos(x)\). The integrating factor is \(\rho=x^2\), giving: $$(x^2y)’=\cos(x)$$ What is the correct general solution for y?

The ODE \(y’+\frac{{4}}{{x}}y=x^3\) has integrating factor \(\rho=x^4\). What is the next line in solving the ODE?

A population is modelled by \(\frac{dP}{dt}=3P(5-P)\) with \(P(0)=P_0\). Find all positive \(P_0\) so that the population is increasing for all t > 0.

We want to solve \(y’+\frac{{3}}{{x}}y=x^2\). The integrating factor is \(\rho=x^3\), giving: $$(x^3y)’=x^5$$ What is the correct general solution for y?

A tank starts with 50 gallons of pure water. Salty water with 3 kg/gal flows in at 2 gal/min; the well-mixed solution leaves at 3 gal/min. After some work you find \(V(t)=50-t\). Let x(t) be the amount of salt (in kg) at time t. Write the ODE for x(t).

Consider \(\frac{dP}{dt}=(P-4)^2(P-10)\). Find all equilibria and classify each.

9. (4 points) List all -values where the function graphed below is not differentiable. Write your answer as “The function is not differentiable when

7. (20 points) Calculate the derivative of the following functions without using limits. Simplify your answer completely.

3. (20 points) Calculate the following limits. Clearly show your work.  If any limits do not exist, explain why not.  “The denominator is zero” is not a sufficient explanation.