Generating...                               quiz03_n14

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

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

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

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

  3. 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

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

  4. Solve the linear ODE $\displaystyle y\,\sec z\,\tan z+{{d}\over{d\,z}}\,y\,\sec z-\cos z=0$.

    $\displaystyle y={{\sin z}\over{\sec z}}+{{c}\over{\sec z}}$ $\displaystyle y=c\,\sin z-\cos z\,\sin z$ $\displaystyle y=\sin ^2z+c\,\sin z$ $\displaystyle y={{c}\over{\sec z}}-{{\cos z}\over{\sec z}}$

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

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


Department of Mathematics
Last modified: 2026-05-20