In each of the Exercises 1 to 10, show that the given differential equation is homogeneous and solve it:
\((x^2+xy)\,dy=(x^2+y^2)\,dx\)
\((x-y)^2=Cx\,e^{-\frac{y}{x}}\)
In each of the Exercises 1 to 10, show that the given differential equation is homogeneous and solve it:
\(y'=\frac{x+y}{x}\)
\(y=x\log|x|+Cx\)
In each of the Exercises 1 to 10, show that the given differential equation is homogeneous and solve it:
\((x-y)\,dy-(x+y)\,dx=0\)
\(\tan^{-1}\left(\frac{y}{x}\right)=-\frac{1}{2}\log(x^2+y^2)+C\)
In each of the Exercises 1 to 10, show that the given differential equation is homogeneous and solve it:
\((x^2-y^2)\,dx+2xy\,dy=0\)
\(x^2+y^2=Cx\)
In each of the Exercises 1 to 10, show that the given differential equation is homogeneous and solve it:
\(x^2\frac{dy}{dx}=x^2-2y^2+xy\)
\(\frac{1}{2\sqrt{2}}\log\left|\frac{x+\sqrt{2}y}{x-\sqrt{2}y}\right|=\log|x|+C\)
In each of the Exercises 1 to 10, show that the given differential equation is homogeneous and solve it:
\(x\,dy-y\,dx=\sqrt{x^2+y^2}\,dx\)
\(y+\sqrt{x^2+y^2}=Cx^2\)
In each of the Exercises 1 to 10, show that the given differential equation is homogeneous and solve it:
\(\{x\cos(\frac{y}{x})+y\sin(\frac{y}{x})\}\,y\,dx=\{y\sin(\frac{y}{x})-x\cos(\frac{y}{x})\}\,x\,dy\)
\(xy\cos\left|\frac{y}{x}\right|=C\)
In each of the Exercises 1 to 10, show that the given differential equation is homogeneous and solve it:
\(x\frac{dy}{dx}-y+x\sin(\frac{y}{x})=0\)
\(x\left[1-\cos\left(\frac{y}{x}\right)\right]=C\sin\left(\frac{y}{x}\right)\)
In each of the Exercises 1 to 10, show that the given differential equation is homogeneous and solve it:
\(y\,dx+x\log\left(\frac{y}{x}\right)\,dy-2x\,dy=0\)
\(cy=\log\left|\frac{y}{x}\right|-1\)
In each of the Exercises 1 to 10, show that the given differential equation is homogeneous and solve it:
\((1+e^{\frac{x}{y}})\,dx+e^{\frac{x}{y}}\left(1-\frac{x}{y}\right)\,dy=0\)
\(ye^{\frac{x}{y}}+x=C\)
For each of the differential equations in Exercises from 11 to 15, find the particular solution satisfying the given condition:
\((x+y)\,dy+(x-y)\,dx=0;\ y=1\ \text{when}\ x=1\)
\(\log(x^2+y^2)+2\tan^{-1}\left(\frac{y}{x}\right)=\frac{\pi}{2}+\log 2\)
For each of the differential equations in Exercises from 11 to 15, find the particular solution satisfying the given condition:
\(x^2\,dy+(xy+y^2)\,dx=0;\ y=1\ \text{when}\ x=1\)
\(y+2x=3x^2y\)
For each of the differential equations in Exercises from 11 to 15, find the particular solution satisfying the given condition:
\([x\sin^2(\frac{y}{x})-y]dx+x\,dy=0;\ y=\frac{\pi}{4}\ \text{when}\ x=1\)
\(\cot\left(\frac{y}{x}\right)=\log|ex|\)
For each of the differential equations in Exercises from 11 to 15, find the particular solution satisfying the given condition:
\(\frac{dy}{dx}-\frac{y}{x}+\cosec(\frac{y}{x})=0;\ y=0\ \text{when}\ x=1\)
\(\cos\left(\frac{y}{x}\right)=\log|ex|\)
For each of the differential equations in Exercises from 11 to 15, find the particular solution satisfying the given condition:
\(2xy+y^2-2x^2\frac{dy}{dx}=0;\ y=2\ \text{when}\ x=1\)
\(y=\frac{2x}{1-\log|x|}\ \ (x\ne 0,\ x\ne e)\)
A homogeneous differential equation of the form \(\frac{dx}{dy}=h\left(\frac{x}{y}\right)\) can be solved by making the substitution.
\(y=vx\)
\(v=yx\)
\(x=vy\)
\(x=v\)
Which of the following is a homogeneous differential equation?
\((4x+6y+5)\,dy-(3y+2x+4)\,dx=0\)
\((xy)\,dx-(x^3+y^3)\,dy=0\)
\((x^3+2y^2)\,dx+2xy\,dy=0\)
\(y^2\,dx+(x^2-xy-y^2)\,dy=0\)