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

Department of Mathematics
Last modified: 2025-03-02