Generating...                               quiz02_n23

  1. Solve the ODE $\displaystyle \left(2\,x\,\cos y-x\,y\,\sin y\right)\,\left({{d}\over{d\,x}}\,y \right)+2\,y\,\cos y=0$ by using the integrating factor $xy$.

    $x^2 y^2 \cos y = c$ $x y \sin y = c$ $x^2 y^2 \sin y = c$ $x y \cos y = c$

  2. Solve the homogeneous ODE $\displaystyle \left(y^3+2\,x^3\right)\,\left({{d}\over{d\,x}}\,y\right)+x^2\,y=0$.

    $\displaystyle {{y^5}\over{5}}+x^3\,y^2=c$ $\displaystyle {{1}\over{3\,y^3}}+{{x^3}\over{3\,y^6}}=c$ $\displaystyle {{x^3}\over{y^2}}-y=c$ $\displaystyle {{y^9}\over{9}}+{{x^3\,y^6}\over{3}}=c$

  3. 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

    $\displaystyle {{y^3}\over{3\,x^3}}-\ln x={{8}\over{3}}$ $\displaystyle {{x^3\,y^3}\over{3}}+{{x^6}\over{6}}={{17}\over{6}}$ $\displaystyle {{x^3\,y^3}\over{3}}+{{x^6}\over{6}}={{1}\over{2}}$ $\displaystyle {{y^3}\over{3\,x^3}}-\ln x={{1}\over{3}}$

  4. Solve the exact ODE $\displaystyle -\sin x\,\sin y\,\left({{d}\over{d\,x}}\,y\right)+\cos x\,\cos y+ \tan x=0$.

    $\displaystyle \sin x\,\cos y+\ln \csc x=c$ $\displaystyle \sin x\,\cos y+\ln \sec x=c$ $\displaystyle \cos x\,\sin y+\ln \sec x=c$ $\displaystyle \cos x\,\sin y+\ln \csc x=c$

  5. 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) = 2

    xy = 1 xy + x = 2 xy + y + x = 5 xy + y = 4


Department of Mathematics
Last modified: 2025-09-14