Generating...                               quiz01_n28

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

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

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

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

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

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


Department of Mathematics
Last modified: 2026-09-10