Generating...                               quiz02_n10

  1. Solve the exact ODE $\displaystyle x\,\left({{d}\over{d\,x}}\,y\right)+y=0$ subject to the initial condition y(1) = 1

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

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

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

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

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

  4. 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

    $\displaystyle {{x^3\,y^3}\over{3}}-{{x^6}\over{6}}={{1}\over{6}}$ $\displaystyle {{y^3}\over{3\,x^3}}+\ln x={{8}\over{3}}$ $\displaystyle {{y^3}\over{3\,x^3}}+\ln x={{1}\over{3}}$ $\displaystyle {{x^3\,y^3}\over{3}}-{{x^6}\over{6}}={{5}\over{2}}$

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

    k = 1 k = −2 k = 2 k = −1


Department of Mathematics
Last modified: 2026-10-28