91Ó°ÊÓ

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. $$ y^{\prime}+3 x^{2} y=0 ; y(x)=C e^{-x^{3}}, y(0)=7 $$

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. $$ e^{y} y^{\prime}=1 ; y(x)=\ln (x+C), y(0)=0 $$

Find explicit particular solutions of the initial value problems in Problems 19 through \(28.\) $$ \frac{d y}{d x}=4 x^{3} y-y, \quad y(1)=-3 $$

$$ y^{\prime}=2 x y+3 x^{2} \exp \left(x^{2}\right), y(0)=5 $$

$$ x y^{\prime}+(2 x-3) y=4 x^{4} $$

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. $$ x \frac{d y}{d x}+3 y=2 x^{5} ; y(x)=\frac{1}{4} x^{5}+C x^{-3}, y(2)=1 $$

Problems 23 and 24 are like Problems 21 and 22, but now use a computer algebra system to plot and print out a slope field for the given differential equation. If you wish (and know how), you can check your manually sketched solution curve by plotting it with the computer: $$ y^{\prime}=x^{2}+y^{2}-1, \quad y(0)=0 ; \quad y(2)=? $$

$$ x y^{\prime}+6 y=3 x y^{4 / 3} $$

Find explicit particular solutions of the initial value problems in Problems 19 through \(28.\) $$ \frac{d y}{d x}+1=2 y, \quad y(1)=1 $$

Find explicit particular solutions of the initial value problems in Problems 19 through \(28.\) $$ \frac{d y}{d x}=y \cot x, \quad y\left(\frac{1}{2} \pi\right)=\frac{1}{2} \pi $$