Generating...                               quiz01_n6

  1. Solve the separable ODE $\displaystyle {{d}\over{d\,x}}\,y-e^{5\,x-4\,y}=0$.

    $\displaystyle -{{e^ {- 4\,y }}\over{4}}-{{e^{5\,x}}\over{5}}=c$ $\displaystyle -e^{4\,y}-e^{5\,x}=c$ $\displaystyle {{e^{4\,y}}\over{4}}-{{e^{5\,x}}\over{5}}=c$ $\displaystyle {{e^{4\,y}}\over{20}}=x+c$

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

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

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

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

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

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

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

    $\displaystyle y=2\,\ln \left(x+5\right)+c$ $\displaystyle y=-3\,\ln \left(x+5\right)+x+c$ $\displaystyle y=-4\,\ln \left(x+7\right)+x+c$ $\displaystyle y=2\,\ln x+x+c$ y = x + c


Department of Mathematics
Last modified: 2025-09-14