Generating...                               quiz03_n28

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

    $\displaystyle y={{x\,\ln x}\over{x+1}}-{{x}\over{x+1}}+{{5}\over{x+1}}$ $\displaystyle y={{4}\over{x+1}}$ $\displaystyle y={{2}\over{x+1}}$ $\displaystyle y={{x\,\ln x}\over{x+1}}-{{x}\over{x+1}}+{{3}\over{x+1}}$

  2. Solve the linear ODE $\displaystyle x\,\left({{d}\over{d\,x}}\,y\right)+y-{{1}\over{x^2}}=0$.

    $\displaystyle y={{\ln x}\over{x^2}}+{{c}\over{x^2}}$ $\displaystyle y={{c}\over{x}}-{{1}\over{x^2}}$ $\displaystyle y={{c}\over{x^2}}-{{1}\over{x^3}}$ $\displaystyle y={{\ln x}\over{x}}+{{c}\over{x}}$

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

    $\displaystyle y={{\sin x}\over{x}}-\cos x+{{c}\over{x}}$ $\displaystyle y={{\sin x}\over{x}}+{{\cos x}\over{x^2}}+{{c}\over{x^2}}$ $\displaystyle y=\sin x+{{\cos x}\over{x}}+{{c}\over{x}}$ $\displaystyle y={{\sin x}\over{x^2}}-{{\cos x}\over{x}}+{{c}\over{x^2}}$

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

    $\displaystyle y={{c}\over{e^{x}-1}}$ $\displaystyle y={{x^2}\over{2\,e^{x}+2}}+{{c}\over{e^{x}+1}}$ $\displaystyle y={{c}\over{e^{x}-1}}-{{x^2}\over{2\,e^{x}-2}}$ $\displaystyle y={{c}\over{e^{x}+1}}$

  5. Solve the Bernoulli ODE $\displaystyle {{d}\over{d\,x}}\,y-x\,y^3+2\,y=0$.

    $\displaystyle y={{e^ {- x }}\over{\left(x\,e^ {- 3\,x }+{{e^ {- 3\,x }}\over{3}}+ c\right)^{{{1}\over{3}}}}}$ $\displaystyle y={{e^ {- 2\,x }}\over{\left({{x\,e^ {- 6\,x }}\over{2}}+{{e^ {- 6 \,x }}\over{12}}+c\right)^{{{1}\over{3}}}}}$ $\displaystyle y={{e^ {- 2\,x }}\over{\sqrt{{{x\,e^ {- 4\,x }}\over{2}}+{{e^ {- 4 \,x }}\over{8}}+c}}}$ $\displaystyle y={{e^ {- x }}\over{\sqrt{x\,e^ {- 2\,x }+{{e^ {- 2\,x }}\over{2}}+ c}}}$


Department of Mathematics
Last modified: 2025-09-14