91Ó°ÊÓ

Solve the differential equation. $$ y^{\prime}=\frac{\sqrt{x}}{3 y} $$

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

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 \(y^{\prime}=4 y\) \(3 y^{\prime}+4 y=e^{-x}\) \(y^{\prime}=2 x y /\left(x^{2}-y^{2}\right)\) \(\frac{d y}{d x}=\frac{x y}{y^{2}-1}\) \(y^{\prime \prime}+y=0\) \(y^{\prime \prime}+2 y^{\prime}+2 y=0\) \(y^{\prime \prime}+y=\tan x\) \(y^{\prime \prime}+2 y^{\prime}=2 e^{x}\)

Solve the first-order linear differential equation. $$ \frac{d y}{d x}+\left(\frac{2}{x}\right) y=3 x+2 $$

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

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

Solve the first-order linear differential equation. $$ y^{\prime}-y=10 $$

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. $$ y^{\prime}=x(1+y) $$

Find the general solution of the differential equation. \(y y^{\prime}=6 \cos (\pi x)\)