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

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

  2. Solve the linear ODE $\displaystyle x^2\,\left({{d}\over{d\,x}}\,y\right)-2\,x\,\left({{d}\over{d\,x}} \,y\right)+2\,y=0$ subject to the initial condition y(3) = 9

    $\displaystyle y={{3\,x}\over{x-2}}$ $\displaystyle y={{2\,x}\over{x-2}}$ $\displaystyle y={{x^3}\over{2\,x-4}}-{{5\,x}\over{2\,x-4}}$ $\displaystyle y={{x^3}\over{2\,x-4}}-{{3\,x}\over{2\,x-4}}$

  3. Solve the Ricatti ODE $\displaystyle {{d}\over{d\,x}}\,y-y^2+{{y}\over{x}}+{{4}\over{x^2}}=0$ using a known solution $\displaystyle y_{1}={{2}\over{x}}$ .

    $\displaystyle y-{{2}\over{x}}={{1}\over{\left(c-{{1}\over{2\,x^2}}\right)\,x^3}}$ $\displaystyle -{{x^2\,\left(y-{{2}\over{x}}\right)+2\,x}\over{2\,\left(y-{{2 }\over{x}}\right)}}=c$ $\displaystyle y-{{2}\over{x}}={{1}\over{\left(c-{{1}\over{4\,x^4}}\right)\,x^5}}$ $\displaystyle y-{{2}\over{x}}={{x^3}\over{c-{{x^4}\over{4}}}}$

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

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

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

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


Department of Mathematics
Last modified: 2026-07-17