Generating...                               quiz03_n19

  1. Solve the linear ODE $\displaystyle -\sin z-y\,\cot z\,\csc z+{{d}\over{d\,z}}\,y\,\csc z=0$.

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

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

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

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

    $\displaystyle y=c\,e^ {- x }+x^2-2\,x+2$ $\displaystyle y=c\,e^ {- x }+2\,x^2-4\,x+4$ $\displaystyle y=c\,e^ {- {{x^3}\over{3}} }+2$ $\displaystyle y=c\,e^ {- {{x^3}\over{3}} }+1$

  4. Solve the linear ODE $\displaystyle x^2\,\left({{d}\over{d\,x}}\,y\right)-2\,x\,\left({{d}\over{d\,x}} \,y\right)+2\,y-x^3=0$ subject to the initial condition y(3) = 6

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

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

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


Department of Mathematics
Last modified: 2025-06-19