Linear equations questions and answers

Recent questions in Second order linear equations

Second order linear equations
ANSWERED
asked 2021-05-29

The integrating factor method, which was an effective method for solving first-order differential equations, is not a viable approach for solving second-order equstions. To see what happens, even for the simplest equation, consider the differential equation \(\displaystyle{y}{''}+{3}{y}'+{2}{y}={f{{\left({t}\right)}}}\). Lagrange sought a function \(\displaystyle\mu{\left({t}\right)}μ{\left({t}\right)}\) such that if one multiplied the left-hand side of \(\displaystyle{y}{''}+{3}{y}'+{2}{y}={f{{\left({t}\right)}}}\) bu \(\displaystyle\mu{\left({t}\right)}μ{\left({t}\right)}\), one would get \(\displaystyle\mu{\left({t}\right)}{\left[{y}{''}+{y}'+{y}\right]}={d}{\frac{{{d}}}{{{\left.{d}{t}\right.}}}}{\left[\mu{\left({t}\right)}{y}+{g{{\left({t}\right)}}}{y}\right]}\) where g(t)g(t) is to be determined. In this way, the given differential equation would be converted to \(\displaystyle{\frac{{{d}}}{{{\left.{d}{t}\right.}}}}{\left[\mu{\left({t}\right)}{y}'+{g{{\left({t}\right)}}}{y}\right]}=\mu{\left({t}\right)}{f{{\left({t}\right)}}}\), which could be integrated, giving the first-order equation \(\displaystyle\mu{\left({t}\right)}{y}'+{g{{\left({t}\right)}}}{y}=\int\mu{\left({t}\right)}{f{{\left({t}\right)}}}{\left.{d}{t}\right.}+{c}\) which could be solved by first-order methods. (a) Differentate the right-hand side of \(\displaystyle\mu{\left({t}\right)}{\left[{y}{''}+{y}'+{y}\right]}={d}{\frac{{{d}}}{{{\left.{d}{t}\right.}}}}{\left[\mu{\left({t}\right)}{y}+{g{{\left({t}\right)}}}{y}\right]}\) and set the coefficients of y,y' and y'' equal to each other to find g(t). (b) Show that the integrating factor \(\displaystyle\mu{\left({t}\right)}μ{\left({t}\right)}\) satisfies the second-order homogeneous equation \(\displaystyle\mu{''}-\mu'+\mu={0}\) called the adjoint equation of \(\displaystyle{y}{''}+{3}{y}'+{2}{y}={f{{\left({t}\right)}}}\). In other words, althought it is possible to find an "integrating factor" for second-order differential equations, to find it one must solve a new second-order equation for the integrating factor μ, which might be every bit as hard as the original equation. (c) Show that the adjoint equation of the general second-order linear equation \(\displaystyle{y}{''}+{p}{\left({t}\right)}{y}'+{q}{\left({t}\right)}{y}={f{{\left({t}\right)}}}\) is the homogeneous equation \(\displaystyle\mu{''}-{p}{\left({t}\right)}\mu'+{\left[{q}{\left({t}\right)}-{p}'{\left({t}\right)}\right]}\mu={0}\).

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