Solve the linear ODE $\displaystyle x\,\left({{d}\over{d\,x}}\,y\right)+y-{{1}\over{x}}=0$ .
$\displaystyle y={{c}\over{x}}-{{1}\over{x^2}}$ $\displaystyle y={{\ln x}\over{x}}+{{c}\over{x}}$ $\displaystyle y={{\ln x}\over{x^2}}+{{c}\over{x^2}}$ $\displaystyle y={{c}\over{x^2}}-{{1}\over{x^3}}$
Solve the linear ODE $\displaystyle {{d}\over{d\,x}}\,y+x^2\,y-x^2=0$ .
$\displaystyle y=c\,e^ {- {{x^3}\over{3}} }+2$ $\displaystyle y=c\,e^ {- x }+x^2-2\,x+2$ $\displaystyle y=c\,e^ {- {{x^3}\over{3}} }+1$ $\displaystyle y=c\,e^ {- x }+2\,x^2-4\,x+4$
Solve the linear ODE $\displaystyle x^2\,\left({{d}\over{d\,x}}\,y\right)+2\,x\,y-x\,\sin x=0$ .
$\displaystyle y=\sin x+{{\cos x}\over{x}}+{{c}\over{x}}$ $\displaystyle y={{\sin x}\over{x^2}}-{{\cos x}\over{x}}+{{c}\over{x^2}}$ $\displaystyle y={{\sin x}\over{x}}-\cos x+{{c}\over{x}}$ $\displaystyle y={{\sin x}\over{x}}+{{\cos x}\over{x^2}}+{{c}\over{x^2}}$
Solve the linear ODE $\displaystyle x^2\,\left({{d}\over{d\,x}}\,y\right)-2\,x\,\left({{d}\over{d\,x}} \,y\right)+2\,y-x^3=0$ subject to the initial condition y(3) = 6
$\displaystyle y={{2\,x}\over{x-2}}$ $\displaystyle y={{3\,x}\over{x-2}}$ $\displaystyle y={{x^3}\over{2\,x-4}}-{{3\,x}\over{2\,x-4}}$ $\displaystyle y={{x^3}\over{2\,x-4}}-{{5\,x}\over{2\,x-4}}$
Solve the linear ODE $\displaystyle y\,\sec z\,\tan z+{{d}\over{d\,z}}\,y\,\sec z-\cos z=0$ .
$\displaystyle y={{\sin z}\over{\sec z}}+{{c}\over{\sec z}}$ $\displaystyle y=c\,\sin z-\cos z\,\sin z$ $\displaystyle y=\sin ^2z+c\,\sin z$ $\displaystyle y={{c}\over{\sec z}}-{{\cos z}\over{\sec z}}$

Department of Mathematics
Last modified: 2026-04-29