Solve the homogeneous ODE $\displaystyle x^2\,y-\left(y^3+2\,x^3\right)\,\left({{d}\over{d\,x}}\,y\right)=0$ .
$\displaystyle {{y^5}\over{5}}+x^3\,y^2=c$ $\displaystyle {{x^3}\over{y^2}}-y=c$ $\displaystyle {{y^9}\over{9}}+{{x^3\,y^6}\over{3}}=c$ $\displaystyle {{1}\over{3\,y^3}}+{{x^3}\over{3\,y^6}}=c$
Solve the homogeneous ODE $\displaystyle x\,\left({{d}\over{d\,x}}\,y\right)-y+x=0$ .
$\displaystyle y=x\,\ln x+c\,x$ $\displaystyle y={{c}\over{x}}-{{x}\over{2}}$ $\displaystyle y=c\,x-x\,\ln x$ $\displaystyle y={{x}\over{2}}+{{c}\over{x}}$
Solve the exact ODE $\displaystyle \left(2\,x^2\,y+2\right)\,\left({{d}\over{d\,x}}\,y\right)+2\,x\,y^ 2=0$ .
$\displaystyle x^2\,y^2+2\,x=c$ $\displaystyle x^2\,y^2+2\,y=c$ $\displaystyle x^2\,y^2=c$ $\displaystyle x^2\,y^2+2\,y+2\,x=c$
Solve the exact ODE $\displaystyle \cos x\,\cos y\,\left({{d}\over{d\,x}}\,y\right)-\sin x\,\sin y+ \tan x=0$ .
$\displaystyle \sin x\,\cos y+\ln \csc x=c$ $\displaystyle \cos x\,\sin y+\ln \csc x=c$ $\displaystyle \cos x\,\sin y+\ln \sec x=c$ $\displaystyle \sin x\,\cos y+\ln \sec x=c$
Solve the homogeneous ODE $\displaystyle x\,y^2\,\left({{d}\over{d\,x}}\,y\right)-y^3-x^3=0$ subject to the initial condition y(1) = 2
$\displaystyle {{y^3}\over{3\,x^3}}-\ln x={{8}\over{3}}$ $\displaystyle {{x^3\,y^3}\over{3}}+{{x^6}\over{6}}={{17}\over{6}}$ $\displaystyle {{x^3\,y^3}\over{3}}+{{x^6}\over{6}}={{1}\over{2}}$ $\displaystyle {{y^3}\over{3\,x^3}}-\ln x={{1}\over{3}}$

Department of Mathematics
Last modified: 2026-04-29