91Ó°ÊÓ

Find the solution \(y(t)\) by recognizing each differential equation as determining unlimited, limited, or logistic growth, and then finding the constants. $$ \begin{array}{l} y^{\prime}=\frac{1}{3} y\left(\frac{1}{2}-y\right) \\ y(0)=\frac{1}{6} \end{array} $$

Solve each differential equation in two ways: first by using an integrating factor, and then by separation of variables. $$ y^{\prime}=x y $$

Find the solution \(y(t)\) by recognizing each differential equation as determining unlimited, limited, or logistic growth, and then finding the constants. $$ \begin{array}{l} y^{\prime}=3 y(10-y) \\ y(0)=20 \end{array} $$

Solve each by the appropriate technique. a. \(y^{\prime}+x y^{2}=0\) b. \(y^{\prime}=y+x^{2} e^{x}\)

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. $$ y^{\prime}=y e^{x}-e^{x} $$

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. $$ y^{\prime}=y e^{x}-y $$

Solve each by the appropriate technique. a. \(y y^{\prime}=x\) b. \(y^{\prime}+y=e^{-x}\)

Find the solution \(y(t)\) by recognizing each differential equation as determining unlimited, limited, or logistic growth, and then finding the constants. $$ \begin{array}{l} y^{\prime}=y(2-y) \\ y(0)=4 \end{array} $$

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. $$ y^{\prime}=a y^{2} \quad \text { (for constant } \left.a>0\right) $$

Find the solution \(y(t)\) by recognizing each differential equation as determining unlimited, limited, or logistic growth, and then finding the constants. $$ \begin{array}{l} y^{\prime}=6 y-2 y^{2} \\ y(0)=1 \end{array} $$