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【General guidance】The answer provided below has been developed in a clear step by step manner.Step1/1ExplanationThe wronskian is the determinant of a matrix whose entries are function and its derivative. A 4th order linear differential equation is \( \mathrm{\frac{{{d}^{{4}}{y}}}{{\left.{d}{x}\right.}^{{4}}}+{P}{\left({x}\right)}\frac{{{d}^{{3}}{y}}}{{\left.{d}{x}\right.}^{{3}}}+{Q}{\left({x}\right)}\frac{{{d}^{{2}}{y}}}{{\left.{d}{x}\right.}^{{2}}}+{R}{\left({x}\right)}\frac{{{\left.{d}{y}\right.}}}{{\left.{d}{x}\right.}}+{S}{\left({x}\right)}{y}={0}} \)Then, the linear differential equation have four linearly independent solution as y_{1}(x) ,y_{2}(x), y_{3}(x) and y_{4}(x).Now the wronskian is \( \begin{align*} \mathrm{{W}{\left({y}_{{{1}}},{y}_{{{2}}},{y}_{{{3}}},{y}_{{{4}}}\right)}{\left({x}\right)}} &= \mathrm{{\left[\begin{matrix}{y}_{{{1}}}{\left({x}\right)}&{y}_{{{2}}}{\left({x}\right)}&{y}_{{{3}}}{\left({x}\right)}&{y}_{{{4}}}{\left({x}\right)}\\\frac{{{\left ... See the full answer