Generating...                               quiz01_n0

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

    $\displaystyle \ln \left(y+1\right)-\ln \left(y-1\right)=x+\ln 4-\ln 2$ $\displaystyle \ln \left(y-1\right)-\ln \left(y+1\right)=x-\ln 4+\ln 2$ $\displaystyle \ln \left(y-1\right)-\ln \left(y+1\right)=x-\ln 5+\ln 3$ $\displaystyle \ln \left(y+1\right)-\ln \left(y-1\right)=x+\ln 5-\ln 3$

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

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

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

    $\displaystyle -{{1}\over{y}}=\arctan e^{x}+c$ $\displaystyle {{1}\over{y}}=\arctan e^{x}+c$ $\displaystyle -{{1}\over{y}}=-{{\ln \left(e^{x}+1\right)}\over{2}}+{{\ln \left( e^{x}-1\right)}\over{2}}+c$ $\displaystyle {{1}\over{y}}=-{{\ln \left(e^{x}+1\right)}\over{2}}+{{\ln \left(e ^{x}-1\right)}\over{2}}+c$

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

    $\displaystyle y=4\,\ln x+x+c$ $\displaystyle y=3\,\ln \left(x+1\right)+x+c$ $\displaystyle y=x+6\,\ln \left(x-1\right)+c$ $\displaystyle y=2\,\ln \left(x+3\right)+x+c$ $\displaystyle y=4\,\ln \left(x+1\right)+c$

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

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


Department of Mathematics
Last modified: 2026-10-28