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Solve the following initial-value problems by using integrating factors. $$ \left(1+x^{2}\right) y^{\prime}=y-1, y(0)=0 $$

You have a cup of coffee at temperature \(70^{\circ} \mathrm{C}\) and the ambient temperature in the room is \(20^{\circ} \mathrm{C}\). Assuming a cooling rate \(k\) of \(0.125\), write and solve the differential equation to describe the temperature of the coffee with respect to time.

Explain your selections. $$ y^{\prime}=y^{2} t^{3} $$

For the following problems, find the general solution to the differential equation.\(y^{\prime}=2 t \sqrt{t^{2}+16}\)

Solve the following initial-value problems by using integrating factors. $$ x y^{\prime}=y+2 x \ln x, y(1)=5 $$

For the following problems, find the general solution to the differential equation.\(x^{\prime}=\operatorname{coth} t+\ln t+3 t^{2}\)

Solve the following initial-value problems by using integrating factors. $$ (2+x) y^{\prime}=y+2+x, y(0)=0 $$

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? $$ y^{\prime}=-3 y, \quad y(0)=1 $$

Solve the following initial-value problems by using integrating factors. $$ y^{\prime}=x y+2 x e^{x}, y(0)=2 $$

For the following problems, find the general solution to the differential equation.\(x^{\prime}=t \sqrt{4+t}\)