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) = 6
$\displaystyle y={{x^3}\over{2\,x-4}}-{{3\,x}\over{2\,x-4}}$ $\displaystyle y={{3\,x}\over{x-2}}$ $\displaystyle y={{2\,x}\over{x-2}}$ $\displaystyle y={{x^3}\over{2\,x-4}}-{{5\,x}\over{2\,x-4}}$
Solve the linear ODE $\displaystyle x\,\left({{d}\over{d\,x}}\,y\right)+y-{{1}\over{x^2}}=0$ .
$\displaystyle y={{\ln x}\over{x^2}}+{{c}\over{x^2}}$ $\displaystyle y={{\ln x}\over{x}}+{{c}\over{x}}$ $\displaystyle y={{c}\over{x^2}}-{{1}\over{x^3}}$ $\displaystyle y={{c}\over{x}}-{{1}\over{x^2}}$
Solve the linear ODE $\displaystyle -y\,\cot z\,\csc z+{{d}\over{d\,z}}\,y\,\csc z-\cos z=0$ .
$\displaystyle y=c\,\sin z-\cos z\,\sin z$ $\displaystyle y=\sin ^2z+c\,\sin z$ $\displaystyle y={{c}\over{\sec z}}-{{\cos z}\over{\sec z}}$ $\displaystyle y={{\sin z}\over{\sec z}}+{{c}\over{\sec z}}$
Solve the linear ODE $\displaystyle x^2\,\left({{d}\over{d\,x}}\,y\right)+2\,x\,y-x\,\sin x=0$ .
$\displaystyle y={{\sin x}\over{x}}+{{\cos x}\over{x^2}}+{{c}\over{x^2}}$ $\displaystyle y={{\sin x}\over{x}}-\cos x+{{c}\over{x}}$ $\displaystyle y={{\sin x}\over{x^2}}-{{\cos x}\over{x}}+{{c}\over{x^2}}$ $\displaystyle y=\sin x+{{\cos x}\over{x}}+{{c}\over{x}}$
Solve the Bernoulli ODE $\displaystyle {{d}\over{d\,x}}\,y-x\,y^3+2\,y=0$ .
$\displaystyle y={{e^ {- x }}\over{\sqrt{x\,e^ {- 2\,x }+{{e^ {- 2\,x }}\over{2}}+ c}}}$ $\displaystyle y={{e^ {- 2\,x }}\over{\sqrt{{{x\,e^ {- 4\,x }}\over{2}}+{{e^ {- 4 \,x }}\over{8}}+c}}}$ $\displaystyle y={{e^ {- x }}\over{\left(x\,e^ {- 3\,x }+{{e^ {- 3\,x }}\over{3}}+ c\right)^{{{1}\over{3}}}}}$ $\displaystyle y={{e^ {- 2\,x }}\over{\left({{x\,e^ {- 6\,x }}\over{2}}+{{e^ {- 6 \,x }}\over{12}}+c\right)^{{{1}\over{3}}}}}$

Department of Mathematics
Last modified: 2025-12-09