For the differential equation, find the general solution:
\(\frac{dy}{dx}+2y=\sin x\)
\(y=\frac{1}{5}(2\sin x-\cos x)+Ce^{-2x}\)
For the differential equation, find the general solution:
\(\frac{dy}{dx}+3y=e^{-2x}\)
\(y=e^{-2x}+Ce^{-3x}\)
For the differential equation, find the general solution:
\(\frac{dy}{dx}+\frac{y}{x}=x^2\)
\(xy=\frac{x^4}{4}+C\)
For the differential equation, find the general solution:
\(\frac{dy}{dx}+(\sec x)y=\tan x\) \((0\le x<\frac{\pi}{2})\)
\(y(\sec x+\tan x)=\sec x+\tan x-x+C\)
For the differential equation, find the general solution:
\(\cos^2 x\,\frac{dy}{dx}+y=\tan x\) \((0\le x<\frac{\pi}{2})\)
\(y=(\tan x-1)+Ce^{-\tan x}\)
For the differential equation, find the general solution:
\(x\frac{dy}{dx}+2y=x^2\log x\)
\(y=\frac{x^2}{16}(4\log|x|-1)+Cx^{-2}\)
For the differential equation, find the general solution:
\(x\log x\,\frac{dy}{dx}+y=\frac{2}{x}\log x\)
\(y\log x=-\frac{2}{x}(1+\log|x|)+C\)
For the differential equation, find the general solution:
\((1+x^2)\,dy+2xy\,dx=\cot x\,dx\) \((x\ne 0)\)
\(y=(1+x^2)^{-1}\log|\sin x|+C(1+x^2)^{-1}\)
For the differential equation, find the general solution:
\(x\frac{dy}{dx}+y-x+xy\cot x=0\) \((x\ne 0)\)
\(y=\frac{1}{x}-\cot x+\frac{C}{x\sin x}\)
For the differential equation, find the general solution:
\((x+y)\frac{dy}{dx}=1\)
\(x+y+1=Ce^{y}\)
For the differential equation, find the general solution:
\(y\,dx+(x-y^2)\,dy=0\)
\(x=\frac{y^2}{3}+\frac{C}{y}\)
For the differential equation, find the general solution:
\((x+3y^2)\frac{dy}{dx}=y\) \((y>0)\)
\(x=3y^2+Cy\)
Find a particular solution satisfying the given condition:
\(\frac{dy}{dx}+2y\tan x=\sin x;\ y=0\ \text{when}\ x=\frac{\pi}{3}\)
\(y=\cos x-2\cos^2 x\)
Find a particular solution satisfying the given condition:
\((1+x^2)\frac{dy}{dx}+2xy=\frac{1}{1+x^2};\ y=0\ \text{when}\ x=1\)
\(y(1+x^2)=\tan^{-1}x-\frac{\pi}{4}\)
Find a particular solution satisfying the given condition:
\(\frac{dy}{dx}-3y\cot x=\sin 2x;\ y=2\ \text{when}\ x=\frac{\pi}{2}\)
\(y=4\sin^3 x-2\sin^2 x\)
Find the equation of a curve passing through the origin given that the slope of the tangent to the curve at any point \((x,y)\) is equal to the sum of the coordinates of the point.
\(x+y+1=e^x\)
Find the equation of a curve passing through the point \((0,2)\) given that the sum of the coordinates of any point on the curve exceeds the magnitude of the slope of the tangent to the curve at that point by 5.
\(y=4-x-2e^x\)
The Integrating Factor of the differential equation \(x\frac{dy}{dx}-y=2x^2\) is
\(e^{-x}\)
\(e^{-y}\)
\(\frac{1}{x}\)
\(x\)
The Integrating Factor of the differential equation \((1-y^2)\frac{dx}{dy}+yx=ay\) \((-1<y<1)\) is
\(\frac{1}{y^2-1}\)
\(\frac{1}{\sqrt{y^2-1}}\)
\(\frac{1}{1-y^2}\)
\(\frac{1}{\sqrt{1-y^2}}\)