91Ó°ÊÓ

Solve the differential equation. $$ \frac{d y}{d x}=y+2 $$

Determine whether the differential equation is linear. Explain your reasoning. $$ y^{\prime}+y \cos x=x y^{2} $$

Solve the differential equation. $$ \frac{d y}{d x}=4-y $$

Verify the solution of the differential equation. Solution 1\. \(y=C e^{4 x}\) 2\. \(y=e^{-x}\) 3\. \(x^{2}+y^{2}=C y\) 4\. \(y^{2}-2 \ln y=x^{2}\) 5\. \(y=C_{1} \cos x+C_{2} \sin x\) 6\. \(y=C_{1} e^{-x} \cos x+C_{2} e^{-x} \sin x\) 7\. \(y=-\cos x \ln |\sec x+\tan x|\) 8\. \(y=\frac{2}{3}\left(e^{-2 x}+e^{x}\right)\) Differential Equation \(y^{\prime}=4 y\) \(3 y^{\prime}+4 y=e^{-x}\) \(y^{\prime}=2 x y /\left(x^{2}-y^{2}\right)\) \(\frac{d y}{d x}=\frac{x y}{y^{2}-1}\) \(y^{\prime \prime}+y=0\) \(y^{\prime \prime}+2 y^{\prime}+2 y=0\) \(y^{\prime \prime}+y=\tan x\) \(y^{\prime \prime}+2 y^{\prime}=2 e^{x}\)

Determine whether the differential equation is linear. Explain your reasoning. Determine whether the differential equation is linear. Explain your reasoning. $$ \frac{1-y^{\prime}}{y}=3 x $$

Find the general solution of the differential equation. \(\frac{d r}{d s}=0.05 s\)

Find the general solution of the differential equation. \((2+x) y^{\prime}=3 y\)

Solve the first-order linear differential equation. $$ \frac{d y}{d x}+\left(\frac{1}{x}\right) y=3 x+4 $$

Verify the solution of the differential equation. Solution 1\. \(y=C e^{4 x}\) 2\. \(y=e^{-x}\) 3\. \(x^{2}+y^{2}=C y\) 4\. \(y^{2}-2 \ln y=x^{2}\) 5\. \(y=C_{1} \cos x+C_{2} \sin x\) 6\. \(y=C_{1} e^{-x} \cos x+C_{2} e^{-x} \sin x\) 7\. \(y=-\cos x \ln |\sec x+\tan x|\) 8\. \(y=\frac{2}{3}\left(e^{-2 x}+e^{x}\right)\) Differential Equation \(y^{\prime}=4 y\) \(3 y^{\prime}+4 y=e^{-x}\) \(y^{\prime}=2 x y /\left(x^{2}-y^{2}\right)\) \(\frac{d y}{d x}=\frac{x y}{y^{2}-1}\) \(y^{\prime \prime}+y=0\) \(y^{\prime \prime}+2 y^{\prime}+2 y=0\) \(y^{\prime \prime}+y=\tan x\) \(y^{\prime \prime}+2 y^{\prime}=2 e^{x}\)

Solve the differential equation. $$ y^{\prime}=\frac{5 x}{y} $$