Eq. (9) with initial conditions (10) can be put in a more general form [29]:
$$\begin{array}{}N[x(t)]=0,\end{array}$$(11)

with *N* an known nonlinear differential operator depending on the unknown function *x*(*t*). The initial conditions are:
$$\begin{array}{}{\displaystyle B\left(x(t),\phantom{\rule{thinmathspace}{0ex}}\frac{dx(t)}{dt}\right)=0.}\end{array}$$(12)

Let *x*_{0}(*t*) be an initial approximation of *x*(*t*). *L* and *B* are arbitrary linear operators such that
$$\begin{array}{}{\displaystyle L\left[{x}_{0}(t)\right]=0,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}B\left({x}_{0}(t),\phantom{\rule{thinmathspace}{0ex}}\frac{d{x}_{0}(t)}{dt}\right)=0.}\end{array}$$(13)

The linear operator *L* is not unique, as in [29].

For *p* ∈ [0, 1] an embedding parameter and *X* an analytic function, we propose to construct a homotopy [29, 30, 31, 32, 33, 34, 35, 36]:
$$\begin{array}{}{\displaystyle \mathcal{H}\left[L\left(X(t,p)\right),\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}H(t,{C}_{i}),\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}N\left(X(t,p)\right)\right],}\\ i=1,2,\dots ,s,\end{array}$$(14)

with properties
$$\begin{array}{}{\displaystyle \mathcal{H}\left[L\left(X(t,0)\right),\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}H(t,{C}_{i}),\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}N\left(X(t,0)\right)\right]=}\\ =L\left(X(t,0)\right)=L\left({x}_{0}(t)\right)=0,\end{array}$$(15)
$$\begin{array}{}{\displaystyle \mathcal{H}\left[L\left(X(t,1)\right),\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}H(t,{C}_{i}),\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}N\left(X(t,1)\right)\right]=}\\ =H(t,{C}_{i})N\left(x(t)\right)=0,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}i=1,2,\dots ,s,\end{array}$$(16)

with *H*(*t*, *C*_{i}) ≠ 0 an arbitrary auxiliary convergence-control function depending on variable *t* and on *s* arbitrary parameters *C*_{1}, *C*_{2}, …, *C*_{s}, unknown now and to be determined later.

Let us consider the function *X* in the form
$$\begin{array}{}{\displaystyle X(t,p)={x}_{0}(t)+p{x}_{1}(t,{C}_{i}).}\end{array}$$(17)

By substituting Eq. (17) into the equation obtained by means of homotopy (14),
$$\begin{array}{}{\displaystyle \mathcal{H}\left[L\left(X(t,p)\right),\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}H(t,{C}_{i}),\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}N\left(X(t,p)\right)\right]=0,}\\ i=1,\dots ,s\end{array}$$(18)

and then equating the coefficients of *p*^{0} and *p*^{1}, we obtain
$$\begin{array}{}{\displaystyle \mathcal{H}\left[L\left(X(t,p)\right),\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}H(t,{C}_{i}),\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}N\left(X(t,p)\right)\right]=}\\ {\displaystyle =L\left({x}_{0}(t)\right)+p\left[L\left({x}_{1}(t,{C}_{i})\right)-L\left({x}_{0}(t)\right)-\right.}\\ \left.-H(t,{C}_{i})N\left({x}_{0}(t)\right)\right]=0,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}i=1,2,\dots ,s.\end{array}$$(19)

From the last equation, we obtain the governing equation of *x*_{0}(*t*) given by Eq. (13) and the governing equation of *x*_{1}(*t*, *C*_{i}), *i*.*e*.:
$$\begin{array}{}{\displaystyle L\left({x}_{1}(t,{C}_{i})\right)=H(t,{C}_{i})N\left({x}_{0}(t)\right),}\\ {\displaystyle B\left({x}_{1}(t,{C}_{i}),\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\frac{d{x}_{1}(t,{C}_{i})}{dt}\right)=0,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}i=1,\dots ,s,}\end{array}$$(20)

where the expression for the nonlinear operator has the form
$$\begin{array}{}{\displaystyle N\left({x}_{0}(t)\right)=\sum _{i=1}^{m}{h}_{i}(t){g}_{i}(t).}\end{array}$$(21)

In Eq. (21), the functions *h*_{i}(*t*) and *g*_{i}(*t*), *i* = 1, …, *m* are known and depend on the initial approximation *x*_{0}(*t*) and also on the nonlinear operator, *m* being a known integer number. In this way, taking into account Eq. (16), from Eq. (17) for *p* = 1, the first-order approximate solution becomes [29]:
$$\begin{array}{}{\displaystyle \overline{x}(t,{C}_{i})={x}_{0}(t)+{x}_{1}(t,{C}_{i}),\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}i=1,\dots ,s.}\end{array}$$(22)

It should be noted that *x*_{0}(*t*) and *x*_{1}(*t*, *C*_{i}) are governed by the linear Eqs. (13) and (20) respectively with initial / boundary conditions that come from the original problem.

Now, we do not solve Eq. (20), but from the theory of differential equations, it is more convenient to consider the unknown function *x*_{1}(*t*, *C*_{i}), as
$$\begin{array}{}{\displaystyle {x}_{1}(t,{C}_{j})=\sum _{i=1}^{n}{H}_{i}(t,{h}_{j}(t),{C}_{j}){g}_{i}(t),\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}j=1,\dots ,s,}\\ {\displaystyle B\left({x}_{1}(t,{C}_{i}),\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\frac{d{x}_{1}(t,{C}_{i})}{dt}\right)=0,}\end{array}$$(23)

where within the expression *H*_{i}(*t*, *h*_{j}(*t*), *C*_{j}) there appear combinations of some functions *h*_{j}, some terms of which are obtained from the corresponding homogeneous equation and the unknown parameters *C*_{j}, *j* = 1, …, *s*, as in [29]. In the sum given by Eq. (23), thenre appear an arbitrary number *n* of such terms.

We cannot demand that *x*_{1}(*t*, *C*_{i}) be solutions of Eq. (20), but *x*(*t*, *C*_{i}) given by Eq. (22) with *x*_{1}(*t*, *C*_{i}) given by Eq. (23) are the solutions of Eq. (11). This is the underlying idea of our method. The convergence of the approximate solution *x*(*t*, *C*_{i}) given by Eq. (22) depends upon the auxiliary functions *H*_{i}(*t*, *h*_{i}, *C*_{j}), *j* = 1, …, *s*. There exist many possibilities for choosing these functions *H*_{i}. We choose the auxiliary functions *H*_{i} so that within Eq. (23), the term $\sum _{i=1}^{n}$ *H*_{i}(*t*, *h*_{j}(*t*), *C*_{j}) *g*_{i}(*t*) has the same form as the term $\sum _{i=1}^{m}$ *h*_{i}(*t*) *g*_{i}(*t*) given by Eq. (21) (see [29]). The first-order approximate solution *x*(*t*, *C*_{i}) also depends on the parameters *C*_{j}, *j* = 1, …, *s*. The values of these parameters can be optimally identified via various methods such as the least-squares method, the Galerkin method, the collocation method, the Ritz method, or by minimizing the square residual error:
$$\begin{array}{}{\displaystyle J({C}_{1},{C}_{2},\dots ,{C}_{s})=\underset{D}{\int}{R}^{2}(t,{C}_{1},{C}_{2},\dots ,{C}_{s})dt,}\end{array}$$(24)

where the residual *R* is given by
$$\begin{array}{}{\displaystyle R(t,{C}_{1},{C}_{2},\dots ,{C}_{s})=N(\overline{x}(t,{C}_{i})).}\end{array}$$(25)

The unknown parameters *C*_{1}, *C*_{2}, …, *C*_{s} can be identified from the conditions
$$\begin{array}{}{\displaystyle \frac{\mathrm{\partial}J}{\mathrm{\partial}{C}_{1}}=\frac{\mathrm{\partial}J}{\mathrm{\partial}{C}_{2}}=\dots =\frac{\mathrm{\partial}J}{\mathrm{\partial}{C}_{s}}=0.}\end{array}$$(26)

Thus, the first-order analytic approximate solution given by Eq. (22) is well-known.

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