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

    $\displaystyle y={{c}\over{x}}-{{x}\over{2}}$ $\displaystyle y=x\,\ln x+c\,x$ $\displaystyle y=c\,x-x\,\ln x$ $\displaystyle y={{x}\over{2}}+{{c}\over{x}}$

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

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

  3. Find the value of k so that the ODE $\displaystyle \left(-2\,x\,e^{2\,y}-8\,x\,y^3\right)\,\left({{d}\over{d\,x}}\,y \right)-e^{2\,y}+k\,y^4+2\,x=0$ becomes exact.

    k = 1 k = −2 k = −1 k = 2

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

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

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

    $\displaystyle x^2\,y^2+2\,x=c$ $\displaystyle x^2\,y^2=c$ $\displaystyle x^2\,y^2+2\,y+2\,x=c$ $\displaystyle x^2\,y^2+2\,y=c$


Department of Mathematics
Last modified: 2026-05-19