Generating...                               quiz02_n17

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

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

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

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

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

  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

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

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

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


Department of Mathematics
Last modified: 2026-09-10