91Ó°ÊÓ

Find the general solution of each differential equation or state that the differential equation is not separable. If the exercise says "and check," verify that your answer is a solution. \(y^{2} y^{\prime}=4 x\)

Integration by parts often involves finding integrals like the following when integrating \(d v\) to find \(v\). Find the following integrals without using integration by parts (using formulas 1 through 7 on the inside back cover). Be ready to find similar integrals during the integration by parts procedure. $$ \int \sqrt{x} d x $$

Determine the type of each differential equation: unlimited growth, limited growth, logistic growth, or none of these. \(y^{\prime}=30(0.5-y)\)

Evaluate each limit (or state that it does not exist). $$ \lim _{x \rightarrow \infty}\left(2-e^{x / 2}\right) $$

Approximate each integral using trapezoidal approximation "by hand" with the given value of \(n\). Round all calculations to three decimal places. \(\int_{0}^{1} \sqrt{1+x^{2}} d x, \quad n=3\)

Approximate each integral using trapezoidal approximation "by hand" with the given value of \(n\). Round all calculations to three decimal places. \(\int_{0}^{1} \sqrt{1+x^{3}} d x, \quad n=3\)

Determine the type of each differential equation: unlimited growth, limited growth, logistic growth, or none of these. \(y^{\prime}=0.4 y(0.01-y)\)

Evaluate each limit (or state that it does not exist). $$ \lim _{x \rightarrow \infty}\left(1-e^{-x / 3}\right) $$

Find the general solution of each differential equation or state that the differential equation is not separable. If the exercise says "and check," verify that your answer is a solution. \(y^{4} y^{\prime}=8 x\)

Integration by parts often involves finding integrals like the following when integrating \(d v\) to find \(v\). Find the following integrals without using integration by parts (using formulas 1 through 7 on the inside back cover). Be ready to find similar integrals during the integration by parts procedure. $$ \int e^{-0.5 t} d t $$