91Ó°ÊÓ

In Problems 17 through 26, first verify that \(y(x)\) satisfies the given differential equation. Then determine a value of the constant \(C\) so that \(y(x)\) satisfies the given initial condition. Use a computer or graphing calculator (if desired) to sketch several typical solutions of the given differential equation, and highlight the one that satisfies the given initial condition. $$ y^{\prime}+y=0 ; y(x)=C e^{-x}, y(0)=2 $$

Find general solutions of the differential equations. Primes denote derivatives with respect to \(x\) throughout. $$ y^{\prime}=(4 x+y)^{2} $$

In Problems, find the position function \(x(t)\) of a moving particle with the given acceleration \(a(t)\), witial position \(x_{0}=x(0)\), and initial velociry \(v_{0}=v(0)\). $$ a(t)=\frac{1}{(t+1)^{3}}, v_{0}=0, x_{0}=0 $$

In Problems 17 through 26, first verify that \(y(x)\) satisfies the given differential equation. Then determine a value of the constant \(C\) so that \(y(x)\) satisfies the given initial condition. Use a computer or graphing calculator (if desired) to sketch several typical solutions of the given differential equation, and highlight the one that satisfies the given initial condition. \(y^{\prime}=2 y ; y(x)=C e^{2 x}, y(0)=3\)

In Problems, find the position function \(x(t)\) of a moving particle with the given acceleration \(a(t)\), witial position \(x_{0}=x(0)\), and initial velociry \(v_{0}=v(0)\). $$ a(t)=50 \sin 5 t, v_{0}=-10, x_{0}=8 $$

Find general solutions of the differential equations. Primes denote derivatives with respect to \(x\) throughout. $$ (x+y) y^{\prime}=1 $$

Find general solutions of the differential equations. Primes denote derivatives with respect to \(x\) throughout. $$ x^{2} y^{\prime}+2 x y=5 y^{3} $$

Find explicit particular solutions of the initial value problems $$ \frac{d y}{d x}=y e^{x}, \quad y(0)=2 e $$

In Problems 17 through 26, first verify that \(y(x)\) satisfies the given differential equation. Then determine a value of the constant \(C\) so that \(y(x)\) satisfies the given initial condition. Use a computer or graphing calculator (if desired) to sketch several typical solutions of the given differential equation, and highlight the one that satisfies the given initial condition. \(y^{\prime}=y+1 ; y(x)=C e^{x}-1, y(0)=5\)

In Problems 17 through 26, first verify that \(y(x)\) satisfies the given differential equation. Then determine a value of the constant \(C\) so that \(y(x)\) satisfies the given initial condition. Use a computer or graphing calculator (if desired) to sketch several typical solutions of the given differential equation, and highlight the one that satisfies the given initial condition. \(y^{\prime}=x-y ; y(x)=C e^{-x}+x-1, y(0)=10\)