91Ó°ÊÓ

Find the solution \(y(t)\) by recognizing each differential equation as determining unlimited, limited, or logistic growth, and then finding the constants. $$ \begin{array}{l} y^{\prime}=6 y \\ y(0)=1.5 \end{array} $$

For each initial value problem: a. Use an Euler's method graphing calculator program to find the estimate for \(y(2)\). Use the interval [0,2] with \(n=50\) segments. b. Solve the differential equation and initial condition exactly by separating variables or using an integrating factor. c. Evaluate the solution that you found in part (b) at \(x=2\). Compare this actual value of \(y(2)\) with the estimate of \(y(2)\) that you found in part (a). $$ \begin{array}{l} \frac{d y}{d x}=0.2 y \\ y(0)=1 \end{array} $$

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. $$ y y^{\prime}=6 x^{2} $$

For each initial value problem: a. Use an Euler's method graphing calculator program to find the estimate for \(y(2)\). Use the interval [0,2] with \(n=50\) segments. b. Solve the differential equation and initial condition exactly by separating variables or using an integrating factor. c. Evaluate the solution that you found in part (b) at \(x=2\). Compare this actual value of \(y(2)\) with the estimate of \(y(2)\) that you found in part (a). $$ \begin{array}{l} \frac{d y}{d x}=-y \\ y(0)=1 \end{array} $$

Solve each first-order linear differential equation. $$ y^{\prime}+\frac{2 x}{x^{2}+1} y=3 $$

Find the solution \(y(t)\) by recognizing each differential equation as determining unlimited, limited, or logistic growth, and then finding the constants. $$ \begin{array}{l} y^{\prime}=0.25 y \\ y(0)=4 \end{array} $$

For each initial value problem: a. Use an Euler's method graphing calculator program to find the estimate for \(y(2)\). Use the interval [0,2] with \(n=50\) segments. b. Solve the differential equation and initial condition exactly by separating variables or using an integrating factor. c. Evaluate the solution that you found in part (b) at \(x=2\). Compare this actual value of \(y(2)\) with the estimate of \(y(2)\) that you found in part (a). $$ \begin{array}{l} y^{\prime}+y=2 e^{x} \\ y(0)=5 \end{array} $$

Find the solution \(y(t)\) by recognizing each differential equation as determining unlimited, limited, or logistic growth, and then finding the constants. $$ \begin{array}{l} y^{\prime}=-y \\ y(0)=100 \end{array} $$

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

Solve each first-order linear differential equation. $$ y^{\prime}-\frac{2}{x} y=6 x^{3}-9 x^{2} $$