91Ó°ÊÓ

Solve each differential equation with the given initial condition. $$ \begin{array}{l} x y^{\prime}+4 y=10 x \\ y(1)=0 \end{array} $$

Use an Euler's method graphing calculator program to find the approximate solution at the stated \(x\) -value, using the given numbers of segments. \(y^{\prime}=x^{2}+y^{2}\) \(y(0)=0\) Approximate the solution at \(x=2\) using: a. \(n=10\) b. \(n=100\) c. \(n=500\)

Find the solution \(y(t)\) by recognizing each differential equation as determining unlimited, limited, or logistic growth, and then finding the constants. $$ \begin{array}{l} y^{\prime}=27-3 y \\ y(0)=0 \end{array} $$

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

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. $$ y^{\prime}=2 \sqrt{y} $$

Find the solution \(y(t)\) by recognizing each differential equation as determining unlimited, limited, or logistic growth, and then finding the constants. $$ \begin{array}{l} y^{\prime}=2-0.01 y \\ y(0)=0 \end{array} $$

Solve each differential equation with the given initial condition. $$ \begin{array}{l} x y^{\prime}=2 y+x^{2} \\ y(1)=3 \end{array} $$

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. $$ y^{\prime}=5+y $$

Solve each differential equation with the given initial condition. $$ \begin{array}{l} x y^{\prime}=3 y+x^{4} \\ y(1)=7 \end{array} $$

Find the solution \(y(t)\) by recognizing each differential equation as determining unlimited, limited, or logistic growth, and then finding the constants. $$ \begin{array}{l} y^{\prime}=6-8 y \\ y(0)=0 \end{array} $$