Generating...                               quiz03_n3

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

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

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

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

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

    $\displaystyle y={{e^ {- x }}\over{\sqrt{x\,e^ {- 2\,x }+{{e^ {- 2\,x }}\over{2}}+ c}}}$ $\displaystyle y={{e^ {- x }}\over{\left(x\,e^ {- 4\,x }+{{e^ {- 4\,x }}\over{4}}+ c\right)^{{{1}\over{4}}}}}$ $\displaystyle y={{e^ {- 2\,x }}\over{\sqrt{{{x\,e^ {- 4\,x }}\over{2}}+{{e^ {- 4 \,x }}\over{8}}+c}}}$ $\displaystyle y={{e^ {- 2\,x }}\over{\left({{x\,e^ {- 8\,x }}\over{2}}+{{e^ {- 8 \,x }}\over{16}}+c\right)^{{{1}\over{4}}}}}$

  5. 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) = 9

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


Department of Mathematics
Last modified: 2025-10-30