4.41.45 $y^{″}(x)(\text{a0}+\text{a1}{\sin }^2(y(x)))+\text{a2}y(x)(\text{a1}{\sin }^2(y(x))+\text{a3})+\text{a1}{y}^{\prime }(x{)}^2+\text{a1}{y}^{\prime }(x{)}^2\sin (y(x))\cos (y(x))=0$

ODE
$$y^{″}(x)(\text{a0}+\text{a1}{\sin }^2(y(x)))+\text{a2}y(x)(\text{a1}{\sin }^2(y(x))+\text{a3})+\text{a1}{y}^{\prime }(x{)}^2+\text{a1}{y}^{\prime }(x{)}^2\sin (y(x))\cos (y(x))=0$$ ODE Classification

[[_2nd_order, _missing_x]]

Book solution method
TO DO

Mathematica ✗
cpu = 600. (sec), leaf count = 0 , timed out

$Aborted

Maple ✓
cpu = 3.671 (sec), leaf count = 818

$$\{{\int }^{y\left(x\right)}((\mathit{a}\mathit{0}+\mathit{a}\mathit{1}){(\tan \left(\mathit{\_}\mathit{h}\right))}^2+\mathit{a}\mathit{0}){\mathrm{e}}^{\mathit{a}\mathit{1}\arctan \left((\mathit{a}\mathit{0}+\mathit{a}\mathit{1})\tan \left(\mathit{\_}\mathit{h}\right)\frac{1}{\sqrt{\mathit{a}\mathit{0}(\mathit{a}\mathit{0}+\mathit{a}\mathit{1})}}\right)\frac{1}{\sqrt{\mathit{a}\mathit{0}(\mathit{a}\mathit{0}+\mathit{a}\mathit{1})}}}\frac{1}{\sqrt{-2(((\int \frac{{(\tan \left(\mathit{\_}\mathit{h}\right))}^2{(\sin \left(\mathit{\_}\mathit{h}\right))}^2\mathit{\_}\mathit{h}}{({(\tan \left(\mathit{\_}\mathit{h}\right))}^2+1)(\mathit{a}\mathit{0}+\mathit{a}\mathit{1}{(\sin \left(\mathit{\_}\mathit{h}\right))}^2)}{\left({\mathrm{e}}^{\frac{\mathit{a}\mathit{1}}{\sqrt{\mathit{a}\mathit{0}(\mathit{a}\mathit{0}+\mathit{a}\mathit{1})}}\arctan \left(\frac{(\mathit{a}\mathit{0}+\mathit{a}\mathit{1})\tan \left(\mathit{\_}\mathit{h}\right)}{\sqrt{\mathit{a}\mathit{0}(\mathit{a}\mathit{0}+\mathit{a}\mathit{1})}}\right)}\right)}^2\mathrm{d}\mathit{\_}\mathit{h}+\int \frac{{(\sin \left(\mathit{\_}\mathit{h}\right))}^2\mathit{\_}\mathit{h}}{({(\tan \left(\mathit{\_}\mathit{h}\right))}^2+1)(\mathit{a}\mathit{0}+\mathit{a}\mathit{1}{(\sin \left(\mathit{\_}\mathit{h}\right))}^2)}{\left({\mathrm{e}}^{\frac{\mathit{a}\mathit{1}}{\sqrt{\mathit{a}\mathit{0}(\mathit{a}\mathit{0}+\mathit{a}\mathit{1})}}\arctan \left(\frac{(\mathit{a}\mathit{0}+\mathit{a}\mathit{1})\tan \left(\mathit{\_}\mathit{h}\right)}{\sqrt{\mathit{a}\mathit{0}(\mathit{a}\mathit{0}+\mathit{a}\mathit{1})}}\right)}\right)}^2\mathrm{d}\mathit{\_}\mathit{h})\mathit{a}\mathit{1}+(\int \frac{{(\tan \left(\mathit{\_}\mathit{h}\right))}^2\mathit{\_}\mathit{h}}{({(\tan \left(\mathit{\_}\mathit{h}\right))}^2+1)(\mathit{a}\mathit{0}+\mathit{a}\mathit{1}{(\sin \left(\mathit{\_}\mathit{h}\right))}^2)}{\left({\mathrm{e}}^{\frac{\mathit{a}\mathit{1}}{\sqrt{\mathit{a}\mathit{0}(\mathit{a}\mathit{0}+\mathit{a}\mathit{1})}}\arctan \left(\frac{(\mathit{a}\mathit{0}+\mathit{a}\mathit{1})\tan \left(\mathit{\_}\mathit{h}\right)}{\sqrt{\mathit{a}\mathit{0}(\mathit{a}\mathit{0}+\mathit{a}\mathit{1})}}\right)}\right)}^2\mathrm{d}\mathit{\_}\mathit{h}+\int \frac{\mathit{\_}\mathit{h}}{({(\tan \left(\mathit{\_}\mathit{h}\right))}^2+1)(\mathit{a}\mathit{0}+\mathit{a}\mathit{1}{(\sin \left(\mathit{\_}\mathit{h}\right))}^2)}{\left({\mathrm{e}}^{\frac{\mathit{a}\mathit{1}}{\sqrt{\mathit{a}\mathit{0}(\mathit{a}\mathit{0}+\mathit{a}\mathit{1})}}\arctan \left(\frac{(\mathit{a}\mathit{0}+\mathit{a}\mathit{1})\tan \left(\mathit{\_}\mathit{h}\right)}{\sqrt{\mathit{a}\mathit{0}(\mathit{a}\mathit{0}+\mathit{a}\mathit{1})}}\right)}\right)}^2\mathrm{d}\mathit{\_}\mathit{h})\mathit{a}\mathit{3})\mathit{a}\mathit{2}\mathit{a}\mathit{0}+{\mathit{a}\mathit{1}}^2\mathit{a}\mathit{2}\int \frac{{(\tan \left(\mathit{\_}\mathit{h}\right))}^2{(\sin \left(\mathit{\_}\mathit{h}\right))}^2\mathit{\_}\mathit{h}}{({(\tan \left(\mathit{\_}\mathit{h}\right))}^2+1)(\mathit{a}\mathit{0}+\mathit{a}\mathit{1}{(\sin \left(\mathit{\_}\mathit{h}\right))}^2)}{\left({\mathrm{e}}^{\frac{\mathit{a}\mathit{1}}{\sqrt{\mathit{a}\mathit{0}(\mathit{a}\mathit{0}+\mathit{a}\mathit{1})}}\arctan \left(\frac{(\mathit{a}\mathit{0}+\mathit{a}\mathit{1})\tan \left(\mathit{\_}\mathit{h}\right)}{\sqrt{\mathit{a}\mathit{0}(\mathit{a}\mathit{0}+\mathit{a}\mathit{1})}}\right)}\right)}^2\mathrm{d}\mathit{\_}\mathit{h}+\mathit{a}\mathit{1}\mathit{a}\mathit{2}\mathit{a}\mathit{3}\int \frac{{(\tan \left(\mathit{\_}\mathit{h}\right))}^2\mathit{\_}\mathit{h}}{({(\tan \left(\mathit{\_}\mathit{h}\right))}^2+1)(\mathit{a}\mathit{0}+\mathit{a}\mathit{1}{(\sin \left(\mathit{\_}\mathit{h}\right))}^2)}{\left({\mathrm{e}}^{\frac{\mathit{a}\mathit{1}}{\sqrt{\mathit{a}\mathit{0}(\mathit{a}\mathit{0}+\mathit{a}\mathit{1})}}\arctan \left(\frac{(\mathit{a}\mathit{0}+\mathit{a}\mathit{1})\tan \left(\mathit{\_}\mathit{h}\right)}{\sqrt{\mathit{a}\mathit{0}(\mathit{a}\mathit{0}+\mathit{a}\mathit{1})}}\right)}\right)}^2\mathrm{d}\mathit{\_}\mathit{h}-\mathit{\_}\mathit{C}\mathit{1}/2)({(\tan \left(\mathit{\_}\mathit{h}\right))}^2+1)((\mathit{a}\mathit{0}+\mathit{a}\mathit{1}){(\tan \left(\mathit{\_}\mathit{h}\right))}^2+\mathit{a}\mathit{0})}}d\mathit{\_}\mathit{h}-x-\mathit{\_}\mathit{C}\mathit{2}=0,{\int }^{y\left(x\right)}-((\mathit{a}\mathit{0}+\mathit{a}\mathit{1}){(\tan \left(\mathit{\_}\mathit{h}\right))}^2+\mathit{a}\mathit{0}){\mathrm{e}}^{\mathit{a}\mathit{1}\arctan \left((\mathit{a}\mathit{0}+\mathit{a}\mathit{1})\tan \left(\mathit{\_}\mathit{h}\right)\frac{1}{\sqrt{\mathit{a}\mathit{0}(\mathit{a}\mathit{0}+\mathit{a}\mathit{1})}}\right)\frac{1}{\sqrt{\mathit{a}\mathit{0}(\mathit{a}\mathit{0}+\mathit{a}\mathit{1})}}}\frac{1}{\sqrt{-2(((\int \frac{{(\tan \left(\mathit{\_}\mathit{h}\right))}^2{(\sin \left(\mathit{\_}\mathit{h}\right))}^2\mathit{\_}\mathit{h}}{({(\tan \left(\mathit{\_}\mathit{h}\right))}^2+1)(\mathit{a}\mathit{0}+\mathit{a}\mathit{1}{(\sin \left(\mathit{\_}\mathit{h}\right))}^2)}{\left({\mathrm{e}}^{\frac{\mathit{a}\mathit{1}}{\sqrt{\mathit{a}\mathit{0}(\mathit{a}\mathit{0}+\mathit{a}\mathit{1})}}\arctan \left(\frac{(\mathit{a}\mathit{0}+\mathit{a}\mathit{1})\tan \left(\mathit{\_}\mathit{h}\right)}{\sqrt{\mathit{a}\mathit{0}(\mathit{a}\mathit{0}+\mathit{a}\mathit{1})}}\right)}\right)}^2\mathrm{d}\mathit{\_}\mathit{h}+\int \frac{{(\sin \left(\mathit{\_}\mathit{h}\right))}^2\mathit{\_}\mathit{h}}{({(\tan \left(\mathit{\_}\mathit{h}\right))}^2+1)(\mathit{a}\mathit{0}+\mathit{a}\mathit{1}{(\sin \left(\mathit{\_}\mathit{h}\right))}^2)}{\left({\mathrm{e}}^{\frac{\mathit{a}\mathit{1}}{\sqrt{\mathit{a}\mathit{0}(\mathit{a}\mathit{0}+\mathit{a}\mathit{1})}}\arctan \left(\frac{(\mathit{a}\mathit{0}+\mathit{a}\mathit{1})\tan \left(\mathit{\_}\mathit{h}\right)}{\sqrt{\mathit{a}\mathit{0}(\mathit{a}\mathit{0}+\mathit{a}\mathit{1})}}\right)}\right)}^2\mathrm{d}\mathit{\_}\mathit{h})\mathit{a}\mathit{1}+(\int \frac{{(\tan \left(\mathit{\_}\mathit{h}\right))}^2\mathit{\_}\mathit{h}}{({(\tan \left(\mathit{\_}\mathit{h}\right))}^2+1)(\mathit{a}\mathit{0}+\mathit{a}\mathit{1}{(\sin \left(\mathit{\_}\mathit{h}\right))}^2)}{\left({\mathrm{e}}^{\frac{\mathit{a}\mathit{1}}{\sqrt{\mathit{a}\mathit{0}(\mathit{a}\mathit{0}+\mathit{a}\mathit{1})}}\arctan \left(\frac{(\mathit{a}\mathit{0}+\mathit{a}\mathit{1})\tan \left(\mathit{\_}\mathit{h}\right)}{\sqrt{\mathit{a}\mathit{0}(\mathit{a}\mathit{0}+\mathit{a}\mathit{1})}}\right)}\right)}^2\mathrm{d}\mathit{\_}\mathit{h}+\int \frac{\mathit{\_}\mathit{h}}{({(\tan \left(\mathit{\_}\mathit{h}\right))}^2+1)(\mathit{a}\mathit{0}+\mathit{a}\mathit{1}{(\sin \left(\mathit{\_}\mathit{h}\right))}^2)}{\left({\mathrm{e}}^{\frac{\mathit{a}\mathit{1}}{\sqrt{\mathit{a}\mathit{0}(\mathit{a}\mathit{0}+\mathit{a}\mathit{1})}}\arctan \left(\frac{(\mathit{a}\mathit{0}+\mathit{a}\mathit{1})\tan \left(\mathit{\_}\mathit{h}\right)}{\sqrt{\mathit{a}\mathit{0}(\mathit{a}\mathit{0}+\mathit{a}\mathit{1})}}\right)}\right)}^2\mathrm{d}\mathit{\_}\mathit{h})\mathit{a}\mathit{3})\mathit{a}\mathit{2}\mathit{a}\mathit{0}+{\mathit{a}\mathit{1}}^2\mathit{a}\mathit{2}\int \frac{{(\tan \left(\mathit{\_}\mathit{h}\right))}^2{(\sin \left(\mathit{\_}\mathit{h}\right))}^2\mathit{\_}\mathit{h}}{({(\tan \left(\mathit{\_}\mathit{h}\right))}^2+1)(\mathit{a}\mathit{0}+\mathit{a}\mathit{1}{(\sin \left(\mathit{\_}\mathit{h}\right))}^2)}{\left({\mathrm{e}}^{\frac{\mathit{a}\mathit{1}}{\sqrt{\mathit{a}\mathit{0}(\mathit{a}\mathit{0}+\mathit{a}\mathit{1})}}\arctan \left(\frac{(\mathit{a}\mathit{0}+\mathit{a}\mathit{1})\tan \left(\mathit{\_}\mathit{h}\right)}{\sqrt{\mathit{a}\mathit{0}(\mathit{a}\mathit{0}+\mathit{a}\mathit{1})}}\right)}\right)}^2\mathrm{d}\mathit{\_}\mathit{h}+\mathit{a}\mathit{1}\mathit{a}\mathit{2}\mathit{a}\mathit{3}\int \frac{{(\tan \left(\mathit{\_}\mathit{h}\right))}^2\mathit{\_}\mathit{h}}{({(\tan \left(\mathit{\_}\mathit{h}\right))}^2+1)(\mathit{a}\mathit{0}+\mathit{a}\mathit{1}{(\sin \left(\mathit{\_}\mathit{h}\right))}^2)}{\left({\mathrm{e}}^{\frac{\mathit{a}\mathit{1}}{\sqrt{\mathit{a}\mathit{0}(\mathit{a}\mathit{0}+\mathit{a}\mathit{1})}}\arctan \left(\frac{(\mathit{a}\mathit{0}+\mathit{a}\mathit{1})\tan \left(\mathit{\_}\mathit{h}\right)}{\sqrt{\mathit{a}\mathit{0}(\mathit{a}\mathit{0}+\mathit{a}\mathit{1})}}\right)}\right)}^2\mathrm{d}\mathit{\_}\mathit{h}-\mathit{\_}\mathit{C}\mathit{1}/2)({(\tan \left(\mathit{\_}\mathit{h}\right))}^2+1)((\mathit{a}\mathit{0}+\mathit{a}\mathit{1}){(\tan \left(\mathit{\_}\mathit{h}\right))}^2+\mathit{a}\mathit{0})}}d\mathit{\_}\mathit{h}-x-\mathit{\_}\mathit{C}\mathit{2}=0\}$$ Mathematica raw input

DSolve[a2*(a3 + a1*Sin[y[x]]^2)*y[x] + a1*y'[x]^2 + a1*Cos[y[x]]*Sin[y[x]]*y'[x]^2 + (a0 + a1*Sin[y[x]]^2)*y''[x] == 0,y[x],x]

Mathematica raw output

$Aborted

Maple raw input

dsolve((a0+a1*sin(y(x))^2)*diff(diff(y(x),x),x)+a1*diff(y(x),x)^2+a1*diff(y(x),x)^2*cos(y(x))*sin(y(x))+a2*y(x)*(a3+a1*sin(y(x))^2) = 0, y(x),'implicit')

Maple raw output

Intat(exp(a1/(a0*(a0+a1))^(1/2)*arctan((a0+a1)*tan(_h)/(a0*(a0+a1))^(1/2)))/(-2*
(((Int(1/(tan(_h)^2+1)*exp(a1/(a0*(a0+a1))^(1/2)*arctan((a0+a1)*tan(_h)/(a0*(a0+
a1))^(1/2)))^2*tan(_h)^2/(a0+a1*sin(_h)^2)*sin(_h)^2*_h,_h)+Int(1/(tan(_h)^2+1)*
exp(a1/(a0*(a0+a1))^(1/2)*arctan((a0+a1)*tan(_h)/(a0*(a0+a1))^(1/2)))^2/(a0+a1*s
in(_h)^2)*sin(_h)^2*_h,_h))*a1+(Int(1/(tan(_h)^2+1)*exp(a1/(a0*(a0+a1))^(1/2)*ar
ctan((a0+a1)*tan(_h)/(a0*(a0+a1))^(1/2)))^2*tan(_h)^2/(a0+a1*sin(_h)^2)*_h,_h)+I
nt(1/(tan(_h)^2+1)*exp(a1/(a0*(a0+a1))^(1/2)*arctan((a0+a1)*tan(_h)/(a0*(a0+a1))
^(1/2)))^2/(a0+a1*sin(_h)^2)*_h,_h))*a3)*a2*a0+a1^2*a2*Int(1/(tan(_h)^2+1)*exp(a
1/(a0*(a0+a1))^(1/2)*arctan((a0+a1)*tan(_h)/(a0*(a0+a1))^(1/2)))^2*tan(_h)^2/(a0
+a1*sin(_h)^2)*sin(_h)^2*_h,_h)+a1*a2*a3*Int(1/(tan(_h)^2+1)*exp(a1/(a0*(a0+a1))
^(1/2)*arctan((a0+a1)*tan(_h)/(a0*(a0+a1))^(1/2)))^2*tan(_h)^2/(a0+a1*sin(_h)^2)
*_h,_h)-1/2*_C1)*(tan(_h)^2+1)*((a0+a1)*tan(_h)^2+a0))^(1/2)*((a0+a1)*tan(_h)^2+
a0),_h = y(x))-x-_C2 = 0, Intat(-exp(a1/(a0*(a0+a1))^(1/2)*arctan((a0+a1)*tan(_h
)/(a0*(a0+a1))^(1/2)))/(-2*(((Int(1/(tan(_h)^2+1)*exp(a1/(a0*(a0+a1))^(1/2)*arct
an((a0+a1)*tan(_h)/(a0*(a0+a1))^(1/2)))^2*tan(_h)^2/(a0+a1*sin(_h)^2)*sin(_h)^2*
_h,_h)+Int(1/(tan(_h)^2+1)*exp(a1/(a0*(a0+a1))^(1/2)*arctan((a0+a1)*tan(_h)/(a0*
(a0+a1))^(1/2)))^2/(a0+a1*sin(_h)^2)*sin(_h)^2*_h,_h))*a1+(Int(1/(tan(_h)^2+1)*e
xp(a1/(a0*(a0+a1))^(1/2)*arctan((a0+a1)*tan(_h)/(a0*(a0+a1))^(1/2)))^2*tan(_h)^2
/(a0+a1*sin(_h)^2)*_h,_h)+Int(1/(tan(_h)^2+1)*exp(a1/(a0*(a0+a1))^(1/2)*arctan((
a0+a1)*tan(_h)/(a0*(a0+a1))^(1/2)))^2/(a0+a1*sin(_h)^2)*_h,_h))*a3)*a2*a0+a1^2*a
2*Int(1/(tan(_h)^2+1)*exp(a1/(a0*(a0+a1))^(1/2)*arctan((a0+a1)*tan(_h)/(a0*(a0+a
1))^(1/2)))^2*tan(_h)^2/(a0+a1*sin(_h)^2)*sin(_h)^2*_h,_h)+a1*a2*a3*Int(1/(tan(_
h)^2+1)*exp(a1/(a0*(a0+a1))^(1/2)*arctan((a0+a1)*tan(_h)/(a0*(a0+a1))^(1/2)))^2*
tan(_h)^2/(a0+a1*sin(_h)^2)*_h,_h)-1/2*_C1)*(tan(_h)^2+1)*((a0+a1)*tan(_h)^2+a0)
)^(1/2)*((a0+a1)*tan(_h)^2+a0),_h = y(x))-x-_C2 = 0