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PDEtools

 dcoeffs
 obtain coefficients of a polynomial differential equation

 Calling Sequence dcoeffs(expr, y(x))

Parameters

 expr - differential expression y(x) - function with respect to which you want to obtain differential coefficients

Description

 • The purpose of this command is to collect the input expression, expr, with respect to the input indeterminate - say y(x) - and its derivatives, and output a sequence of the coefficients of the collected form. The input expression is necessarily polynomial in the indeterminate and its derivatives.
 • Unlike coeffs, the coefficient sequence is returned in a specific order. The output is from higher to lower differential order.
 • If the collected expr is polynomial in y(x) and its derivatives, then the expression is also collected with respect to differing degrees of the same derivative, and the polynomial degree is used to break differential order ties in the ordered output, from higher to lower polynomial degree.

Examples

Consider the following expression.

 > $\mathrm{with}\left(\mathrm{PDEtools}\right):$
 > $\mathrm{diff}\left(y\left(x\right),x,x,x\right)+a{\mathrm{diff}\left(y\left(x\right),x,x\right)}^{2}+b\mathrm{diff}\left(y\left(x\right),x,x\right)+c\mathrm{diff}\left(y\left(x\right),x\right)+dy\left(x\right)+e$
 $\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}{a}{}{\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}^{{2}}{+}{b}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{c}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{d}{}{y}{}\left({x}\right){+}{e}$ (1)

Obtain the coefficients with respect to y(x) and derivatives.

 > $\mathrm{dcoeffs}\left(,y\left(x\right)\right)$
 ${1}{,}{a}{,}{b}{,}{c}{,}{d}{,}{e}$ (2)
 > $\mathrm{expr}≔ax\left(t\right){\mathrm{diff}\left(y\left(t\right),t\right)}^{2}+x\left(t\right)\mathrm{diff}\left(y\left(t\right),t\right)+by\left(t\right)\mathrm{diff}\left(y\left(t\right),t\right)+c{y\left(t\right)}^{2}\mathrm{diff}\left(x\left(t\right),t\right)$
 ${\mathrm{expr}}{≔}{a}{}{x}{}\left({t}\right){}{\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\right)}^{{2}}{+}{x}{}\left({t}\right){}\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\right){+}{b}{}{y}{}\left({t}\right){}\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\right){+}{c}{}{{y}{}\left({t}\right)}^{{2}}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right)\right)$ (3)
 > $\mathrm{dcoeffs}\left(\mathrm{expr},y\left(t\right)\right)$
 ${a}{}{x}{}\left({t}\right){,}{b}{,}{x}{}\left({t}\right){,}{c}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}{}\left({t}\right)\right)$ (4)
 > $\mathrm{dcoeffs}\left(\mathrm{expr},x\left(t\right)\right)$
 ${c}{}{{y}{}\left({t}\right)}^{{2}}{,}{a}{}{\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\right)}^{{2}}{+}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right){,}{b}{}{y}{}\left({t}\right){}\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({t}\right)\right)$ (5)