Generating...                               quiz01_n26

  1. Solve the separable ODE $\displaystyle \left(x-1\right)\,\left({{d}\over{d\,x}}\,y\right)-x-4=0$.

    $\displaystyle y=x+5\,\ln \left(x-1\right)+c$ $\displaystyle y=4\,\ln \left(x+1\right)+x+c$ $\displaystyle y=4\,\ln \left(x-1\right)+c$ $\displaystyle y=x+8\,\ln \left(x-3\right)+c$ $\displaystyle y=4\,\ln x+x+c$

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

    $\displaystyle \cos y={{\sin \left(2\,x\right)}\over{4}}+{{x}\over{2}}+c$ $\displaystyle -\sin y=-{{\sin \left(2\,x\right)}\over{4}}+{{x}\over{2}}+c$ $\displaystyle {{\sin \left(2\,y\right)}\over{4}}-{{y}\over{2}}=\sin x+c$ $\displaystyle -{{\sin \left(2\,y\right)}\over{4}}-{{y}\over{2}}=c-\cos x$

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

    $\displaystyle y=c\,x^2$ y = cx $\displaystyle y=2\,\ln x+c$ $\displaystyle y=\left(c-{{2}\over{x}}\right)\,x$

  4. Solve the separable ODE $\displaystyle y\,\left({{d}\over{d\,x}}\,y\right)-2\,x^2\,\sqrt{y^2+1}=0$ subject to the initial condition y(1) = 1

    $\displaystyle {{\sqrt{y^2+1}}\over{2}}={{x^3}\over{3}}+{{1}\over{\sqrt{2}}}-{{1 }\over{3}}$ $\displaystyle {{\sqrt{y^2+1}}\over{2}}={{x^2}\over{2}}+{{\sqrt{5}}\over{2}}-{{1 }\over{2}}$ $\displaystyle {{\sqrt{y^2+1}}\over{2}}={{x^2}\over{2}}+{{1}\over{\sqrt{2}}}-{{1 }\over{2}}$ $\displaystyle {{\sqrt{y^2+1}}\over{2}}={{x^3}\over{3}}+{{\sqrt{5}}\over{2}}-{{1 }\over{3}}$

  5. Solve the separable ODE $\displaystyle 4\,\left({{d}\over{d\,x}}\,y\right)+y^2-4=0$ subject to the initial condition y(1) = 3

    $\displaystyle \ln \left(y-2\right)-\ln \left(y+2\right)=x-\ln 6+\ln 2-1$ $\displaystyle \ln \left(y+2\right)-\ln \left(y-2\right)=x+\ln 6-\ln 2-1$ $\displaystyle \ln \left(y-2\right)-\ln \left(y+2\right)=x-\ln 5-1$ $\displaystyle \ln \left(y+2\right)-\ln \left(y-2\right)=x+\ln 5-1$


Department of Mathematics
Last modified: 2026-05-20