Quiz

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

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

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

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

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

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

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

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

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

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