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. $$ y^{\prime}=x^{3} y $$

Find the solution \(y(t)\) by recognizing each differential equation as determining unlimited, limited, or logistic growth, and then finding the constants. $$ \begin{array}{l} y^{\prime}=0.05(0.25-y) \\ y(0)=0 \end{array} $$

Solve each differential equation with the given initial condition. $$ \begin{array}{l} y^{\prime}+3 y=12 e^{x} \\ y(0)=5 \end{array} $$

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

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. $$ y^{\prime}=\frac{x}{y} $$

Solve each differential equation with the given initial condition. $$ \begin{array}{l} y^{\prime}+4 y=e^{-3 x} \\ y(0)=4 \end{array} $$

For each initial value problem, use an Euler's method graphing calculator program to find the approximate solution at the stated \(x\) -value, using 50 segments. [Hint: Use an interval that begins at the initial \(x\) -value and ends at the stated \(x\) -value. \(\frac{d y}{d x}=(x-y)^{2}\) \(y(2)=0\) Approximate the solution at \(x=2.8\)

Find the solution \(y(t)\) by recognizing each differential equation as determining unlimited, limited, or logistic growth, and then finding the constants. $$ \begin{array}{l} y^{\prime}=80-2 y \\ y(0)=0 \end{array} $$

Solve each differential equation with the given initial condition. $$ \begin{array}{l} x y^{\prime}+2 y=14 x^{5} \\ y(1)=1 \end{array} $$

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. $$ y^{\prime}=x^{m} y^{n} \quad \text { (for } \left.m>0, n \neq 1\right) $$