The dsolve command uses several methods when trying to find a series solution to an ODE or a system of ODEs. When initial conditions or an expansion point are given, the series is calculated at the given point; otherwise, the series is calculated at the origin.

The first method used is a Newton iteration based on a paper of Keith Geddes. See the References section in this help page.

The second method involves a direct substitution to generate a system of equations, which may be solvable (by solve) to give a series.

The third method is the method of Frobenius for nth order linear DEs. See the References section in this help page.

If the aforementioned methods fail, the function invokes LinearFunctionalSystems[SeriesSolution].

$\mathrm{ode}≔\mathrm{diff}\left(y\left(t\right)\,t\,t\right)+{\mathrm{diff}\left(y\left(t\right)\,t\right)}^2=0$

$\mathrm{ode}≔\frac{{\ⅆ}^2}{\ⅆt^2}y\left(t\right)+{\left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)}^2=0$

$\mathrm{ans}≔\mathrm{dsolve}\left(\left\{\mathrm{ode}\right\}\,y\left(t\right)\,\mathrm{type}='\mathrm{series}'\right)$

$\mathrm{ans}≔y\left(t\right)=y\left(0\right)+\mathrm{D}\left(y\right)\left(0\right)t-\frac{1}{2}{\mathrm{D}\left(y\right)\left(0\right)}^2{t}^2+\frac{1}{3}{\mathrm{D}\left(y\right)\left(0\right)}^3{t}^3-\frac{1}{4}{\mathrm{D}\left(y\right)\left(0\right)}^4{t}^4+\frac{1}{5}{\mathrm{D}\left(y\right)\left(0\right)}^5{t}^5+O\left(t^6\right)$

$\mathrm{ans}≔\mathrm{dsolve}\left(\left\{\mathrm{ode}\,y\left(a\right)=\mathrm{Y\_a}\,\mathrm{D}\left(y\right)\left(a\right)=\mathrm{DY\_a}\right\}\,y\left(t\right)\,\mathrm{type}='\mathrm{series}'\right)$

$\mathrm{ans}≔y\left(t\right)=\mathrm{Y\_a}+\mathrm{DY\_a}\left(t-a\right)-\frac{1}{2}{\mathrm{DY\_a}}^2{\left(t-a\right)}^2+\frac{1}{3}{\mathrm{DY\_a}}^3{\left(t-a\right)}^3-\frac{1}{4}{\mathrm{DY\_a}}^4{\left(t-a\right)}^4+\frac{1}{5}{\mathrm{DY\_a}}^5{\left(t-a\right)}^5+O\left({\left(t-a\right)}^6\right)$

$\mathrm{ans}≔\mathrm{dsolve}\left(\left(1-t^2\right)\mathrm{diff}\left(y\left(t\right)\,t\,t\right)-2ty\left(t\right)-y\left(t\right)\,y\left(t\right)\,'\mathrm{series}'\,t=1\right)$

$\mathrm{ans}≔y\left(t\right)=\mathrm{\_C1}\left(t-1\right)\left(1-\frac{3}{4}\left(t-1\right)+\frac{7}{48}{\left(t-1\right)}^2+\frac{1}{128}{\left(t-1\right)}^3-\frac{157}{15360}{\left(t-1\right)}^4+\frac{3371}{921600}{\left(t-1\right)}^5+O\left({\left(t-1\right)}^6\right)\right)+\mathrm{\_C2}\left(\ln \left(t-1\right)\left(-\frac{3}{2}\left(t-1\right)+\frac{9}{8}{\left(t-1\right)}^2-\frac{7}{32}{\left(t-1\right)}^3-\frac{3}{256}{\left(t-1\right)}^4+\frac{157}{10240}{\left(t-1\right)}^5+O\left({\left(t-1\right)}^6\right)\right)+\left(1-\frac{29}{16}{\left(t-1\right)}^2+\frac{21}{32}{\left(t-1\right)}^3-\frac{131}{3072}{\left(t-1\right)}^4-\frac{2219}{102400}{\left(t-1\right)}^5+O\left({\left(t-1\right)}^6\right)\right)\right)$

$\mathrm{Order}≔3$

$\mathrm{Order}≔3$

$\mathrm{sys}≔\left\{\mathrm{diff}\left(x\left(t\right)\,t\right)=y\left(t\right)\,\mathrm{diff}\left(y\left(t\right)\,t\right)=-x\left(t\right)\right\}$

$\mathrm{sys}≔\left\{\frac{\ⅆ}{\ⅆt}x\left(t\right)=y\left(t\right)\,\frac{\ⅆ}{\ⅆt}y\left(t\right)=-x\left(t\right)\right\}$

$\mathrm{ans}≔\mathrm{dsolve}\left(\mathrm{sys}\mathbf{union}\left\{x\left(0\right)=A\,y\left(0\right)=B\right\}\,\left\{x\left(t\right)\,y\left(t\right)\right\}\,\mathrm{type}='\mathrm{series}'\right)$

$\mathrm{ans}≔\left\{x\left(t\right)=A+Bt-\frac{1}{2}A{t}^2+O\left(t^3\right)\,y\left(t\right)=B-At-\frac{1}{2}B{t}^2+O\left(t^3\right)\right\}$

$\mathrm{sys}≔\left[\mathrm{diff}\left(\mathrm{y1}\left(x\right)\,x\right)-\mathrm{y1}\left(x\right)+x\mathrm{y2}\left(x\right)=x^3\,x\mathrm{diff}\left(\mathrm{y2}\left(x\right)\,x\right)-2\mathrm{y2}\left(x\right)\right]$

$\mathrm{sys}≔\left[\frac{\ⅆ}{\ⅆx}\mathrm{y1}\left(x\right)-\mathrm{y1}\left(x\right)+x\mathrm{y2}\left(x\right)=x^3\,x\left(\frac{\ⅆ}{\ⅆx}\mathrm{y2}\left(x\right)\right)-2\mathrm{y2}\left(x\right)\right]$

$\mathrm{vars}≔\left[\mathrm{y1}\left(x\right)\,\mathrm{y2}\left(x\right)\right]$

$\mathrm{vars}≔\left[\mathrm{y1}\left(x\right)\,\mathrm{y2}\left(x\right)\right]$

$\mathrm{dsolve}\left(\left\{\mathrm{op}\left(\mathrm{sys}\right)\right\}\mathbf{union}\left\{\mathrm{y1}\left(0\right)=13\right\}\,\mathrm{vars}\,'\mathrm{series}'\right)$

$\left\{\mathrm{y1}\left(x\right)=13+13x+\frac{13}{2}{x}^2+O\left(x^3\right)\,\mathrm{y2}\left(x\right)=\frac{{\mathrm{D}}^{\left(2\right)}\left(\mathrm{y2}\right)\left(0\right)}{2}{x}^2+O\left(x^3\right)\right\}$

Forsyth, A.R. Theory of Differential Equations. Cambridge: University Press, 1906. pp. 78-90

Geddes, Keith.  "Convergence Behaviour of the Newton Iteration for First Order Differential Equations". Proceedings of EUROSAM '79. pp.189-199.

Ince, E.L. Ordinary Differential Equations. Dover Publications, 1956. pp. 398-406.