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Estimate the following solutions using Euler's method with \(n=5\) steps over the interval \(t=[0,1] .\) If you are able to solve the initialvalue problem exactly, compare your solution with the exact solution. If you are unable to solve the initial-value problem, the exact solution will be provided for you to compare with Euler's method. How accurate is Euler's method? $$ y^{\prime}=e^{(x+y)}, y(0)=-1 . \text { Exact solution is } y=-\ln \left(e+1-e^{x}\right) $$

Solve the following initial-value problems starting from \(y(t=0)=1\) and \(y(t=0)=-1 .\) Draw both solutions on the same graph\(\frac{d y}{d t}=-t\)

Assume an initial nutrient amount of \(I\) kilograms in a tank with \(L\) liters. Assume a concentration of \(c \mathrm{~kg} / \mathrm{L}\) being pumped in at a rate of \(r \mathrm{~L} / \mathrm{min}\). The tank is well mixed and is drained at a rate of \(r \mathrm{~L} / \mathrm{min}\). Find the equation describing the amount of nutrient in the tank.

Estimate the following solutions using Euler's method with \(n=5\) steps over the interval \(t=[0,1] .\) If you are able to solve the initialvalue problem exactly, compare your solution with the exact solution. If you are unable to solve the initial-value problem, the exact solution will be provided for you to compare with Euler's method. How accurate is Euler's method? $$ y^{\prime}=y^{2} \ln (x+1), y(0)=1 . \text { Exact solution is } y=-\frac{1}{(x+1)(\ln (x+1)-1)} $$

Solve the following initial-value problems starting from \(y(t=0)=1\) and \(y(t=0)=-1 .\) Draw both solutions on the same graph\(\frac{d y}{d t}=2 y\)

Leaves accumulate on the forest floor at a rate of \(2 \mathrm{~g} / \mathrm{cm}^{2} / \mathrm{yr}\) and also decompose at a rate of \(90 \%\) per year. Write a differential equation governing the number of grams of leaf litter per square centimeter of forest floor, assuming at time 0 there is no leaf litter on the ground. Does this amount approach a steady value? What is that value?

Estimate the following solutions using Euler's method with \(n=5\) steps over the interval \(t=[0,1] .\) If you are able to solve the initialvalue problem exactly, compare your solution with the exact solution. If you are unable to solve the initial-value problem, the exact solution will be provided for you to compare with Euler's method. How accurate is Euler's method? $$ y^{\prime}=2^{x}, y(0)=0, \text { Exact solution is } y=\frac{2^{x}-1}{\ln (2)} $$

Leaves accumulate on the forest floor at a rate of \(4 \mathrm{~g} / \mathrm{cm}^{2} / \mathrm{yr}\). These leaves decompose at a rate of \(10 \%\) per year. Write a differential equation governing the number of grams of leaf litter per square centimeter of forest floor. Does this amount approach a steady value? What is that value?

Solve the following initial-value problems starting from \(y(t=0)=1\) and \(y(t=0)=-1 .\) Draw both solutions on the same graph\(\frac{d y}{d t}=-y\)

Solve the following initial-value problems starting from \(y(t=0)=1\) and \(y(t=0)=-1 .\) Draw both solutions on the same graph\(\frac{d y}{d t}=2\)