Reduction of order

The reduction of order technique allows one to find a second solution to a differential equation when one solution is already found.

Take a linear second order constant coefficient homogeneous ODE (see Second order linear ODE) with one repeated-root solution. Add u(t) as follows

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(t)=u(t)e^{2t} }

Substituting into the ODE, and you get

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u'' = 0 }

Integrating up, and you get u in the form

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u=te^{2t} }