When the input ODE is a homogeneous linear ode with polynomial coefficients, and the optional arguments 'formal_solution' (or 'type=formal_solution') and 'coeffs'=coeff_type are given, the dsolve command returns a set of formal solutions with the specified coefficients at the given point (the default is at the origin). For more information, see Slode[mhypergeom_formal_sol] and Slode[dAlembertian_formal_sol].

$\mathrm{ode}≔\left(x^2+1\right)x\mathrm{diff}\left(y\left(x\right)\,x\,x\,x\right)+3\left(2x^2+1\right)\mathrm{diff}\left(y\left(x\right)\,x\,x\right)-12y\left(x\right)$

$\mathrm{ode}≔\left(x^2+1\right)x\left(\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)\right)+3\left(2x^2+1\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)-12y\left(x\right)$

$\mathrm{dsolve}\left(\mathrm{ode}\,y\left(x\right)\,'\mathrm{formal\_solution}'\,'\mathrm{coeffs}'='\mathrm{mhypergeom}'\right)$

$y\left(x\right)=\frac{\left(2x^3+x\right)\mathrm{\_C1}+\frac{\mathrm{\_C2}\left(\sum _{\mathrm{\_n}=1}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(\mathrm{\_n}-\frac{3}{2}\right){\left(\mathrm{-1}\right)}^{\mathrm{\_n}}{x}^{2\mathrm{\_n}}}{\mathrm{\Gamma }\left(\mathrm{\_n}\right)}\right)}{2\sqrt{\mathrm{\pi }}}}{x}$

$\mathrm{ode}≔\left(-4-x^2+2x\right)y\left(x\right)+\left(2x-3x^3-x^2\right)\mathrm{diff}\left(y\left(x\right)\,x\right)+\left(x^3-x^4\right)\mathrm{diff}\left(y\left(x\right)\,\mathrm{`\$`}\left(x\,2\right)\right)$

$\mathrm{ode}≔\left(-x^2+2x-4\right)y\left(x\right)+\left(-3x^3-x^2+2x\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+\left(-x^4+x^3\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)$

$\mathrm{dsolve}\left(\mathrm{ode}\,y\left(x\right)\,'\mathrm{formal\_solution}'\,'\mathrm{coeffs}'='\mathrm{dAlembertian}'\right)$

$y\left(x\right)=x^2\left(-\frac{\left(\sum _{\mathrm{\_n}=0}^{\mathrm{\infty }}{x}^{\mathrm{\_n}}\right)}{2}+\left(\sum _{\mathrm{\_n}=0}^{\mathrm{\infty }}\left(\sum _{\mathrm{\_n1}=0}^{\mathrm{\_n}-1}\frac{{\left(-\frac{1}{2}\right)}^{\mathrm{\_n1}}}{\prod _{\mathrm{\_k}=0}^{\mathrm{\_n1}-1}\frac{\mathrm{\_k}+2}{{\left(\mathrm{\_k}+3\right)}^2}}\right)x^{\mathrm{\_n}}\right)\right)\mathrm{\_C1}+\frac{{\ⅇ}^{\frac{2}{x}}\left(\left(\sum _{\mathrm{\_n}=0}^{\mathrm{\infty }}{x}^{\mathrm{\_n}}\right)-\frac{1}{3}\right)\mathrm{\_C2}}{x}$

$\mathrm{ode}≔-\left(x-1\right)y\left(x\right)-\left(2x^2-4x-1\right)\mathrm{diff}\left(y\left(x\right)\,x\right)-\frac{1}{2}x\left(x+1\right)\left(x-6\right)\mathrm{diff}\left(y\left(x\right)\,\mathrm{`\$`}\left(x\,2\right)\right)+\frac{1}{2}\left(2+x\right)x^2\mathrm{diff}\left(y\left(x\right)\,\mathrm{`\$`}\left(x\,3\right)\right)$

$\mathrm{ode}≔-\left(x-1\right)y\left(x\right)-\left(2x^2-4x-1\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)-\frac{x\left(x+1\right)\left(x-6\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)}{2}+\frac{\left(x+2\right)x^2\left(\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)\right)}{2}$

$\mathrm{dsolve}\left(\mathrm{ode}\,y\left(x\right)\,'\mathrm{formal\_solution}'\,'\mathrm{coeffs}'='\mathrm{dAlembertian}'\,'\mathrm{point}'=a\right)$

$y\left(x\right)=\mathrm{\_C1}\left(\sum _{\mathrm{\_n}=0}^{\mathrm{\infty }}\frac{{\left(-\frac{1}{a+2}\right)}^{\mathrm{\_n}}{\left(x-a\right)}^{\mathrm{\_n}}}{\prod _{\mathrm{\_k}=0}^{\mathrm{\_n}-1}\frac{\mathrm{\_k}+1}{\mathrm{\_k}+2}}\right)+\mathrm{\_C2}\left(-a-2\right)\left(\sum _{\mathrm{\_n}=0}^{\mathrm{\infty }}\frac{{\left(-\frac{1}{a+2}\right)}^{\mathrm{\_n}}\left(\sum _{\mathrm{\_n1}=0}^{\mathrm{\_n}-1}\frac{\left(\mathrm{\_n1}+1\right){\left(\prod _{\mathrm{\_k}=0}^{\mathrm{\_n1}-1}\frac{\mathrm{\_k}+1}{\mathrm{\_k}+2}\right)}^2{\left(\frac{a+2}{a}\right)}^{\mathrm{\_n1}}}{\mathrm{\_n1}+2}\right){\left(x-a\right)}^{\mathrm{\_n}}}{\prod _{\mathrm{\_k}=0}^{\mathrm{\_n}-1}\frac{\mathrm{\_k}+1}{\mathrm{\_k}+2}}\right)+\left(3a\left(a+2\right)\left(\sum _{\mathrm{\_n}=0}^{\mathrm{\infty }}\frac{{\left(-\frac{1}{a+2}\right)}^{\mathrm{\_n}}\left(\sum _{\mathrm{\_n1}=0}^{\mathrm{\_n}-1}\frac{\left(\mathrm{\_n1}+1\right){\left(\prod _{\mathrm{\_k}=0}^{\mathrm{\_n1}-1}\frac{\mathrm{\_k}+1}{\mathrm{\_k}+2}\right)}^2{\left(\frac{a+2}{a}\right)}^{\mathrm{\_n1}}\left(\sum _{\mathrm{\_n2}=0}^{\mathrm{\_n1}-1}\frac{\left(\mathrm{\_n2}+2\right)\left(\prod _{\mathrm{\_k}=0}^{\mathrm{\_n2}-1}\frac{\mathrm{\_k}+2}{\mathrm{\_k}+3}\right)}{\mathrm{\_n2}+1}\right)}{\mathrm{\_n1}+2}\right){\left(x-a\right)}^{\mathrm{\_n}}}{\prod _{\mathrm{\_k}=0}^{\mathrm{\_n}-1}\frac{\mathrm{\_k}+1}{\mathrm{\_k}+2}}\right)-a^2\left(a+2\right)\left(\sum _{\mathrm{\_n}=0}^{\mathrm{\infty }}\frac{{\left(-\frac{1}{a+2}\right)}^{\mathrm{\_n}}\left(\sum _{\mathrm{\_n1}=0}^{\mathrm{\_n}-1}\frac{\left(\mathrm{\_n1}+1\right){\left(\prod _{\mathrm{\_k}=0}^{\mathrm{\_n1}-1}\frac{\mathrm{\_k}+1}{\mathrm{\_k}+2}\right)}^2{\left(\frac{a+2}{a}\right)}^{\mathrm{\_n1}}\left(\sum _{\mathrm{\_n2}=0}^{\mathrm{\_n1}-1}\frac{\left(\mathrm{\_n2}+2\right)\left(\prod _{\mathrm{\_k}=0}^{\mathrm{\_n2}-1}\frac{\mathrm{\_k}+2}{\mathrm{\_k}+3}\right)\left(\sum _{\mathrm{\_n3}=0}^{\mathrm{\_n2}-1}\frac{\left(\mathrm{\_n3}+3\right){\left(-a\right)}^{\mathrm{\_n3}}}{\left(\mathrm{\_n3}+1\right)\left(\prod _{\mathrm{\_k}=0}^{\mathrm{\_n3}-1}\frac{\left(\mathrm{\_k}+4\right)\left(\mathrm{\_k}+1\right)}{\mathrm{\_k}+3}\right)}\right)}{\mathrm{\_n2}+1}\right)}{\mathrm{\_n1}+2}\right){\left(x-a\right)}^{\mathrm{\_n}}}{\prod _{\mathrm{\_k}=0}^{\mathrm{\_n}-1}\frac{\mathrm{\_k}+1}{\mathrm{\_k}+2}}\right)\right)\mathrm{\_C3}$