y′′(x)( a0 + a1sin2(y(x))) + a2y(x)(a1sin2(y(x)) + a3) + a1y′(x)2 + a1y′(x)2 sin(y(x))cos(y(x)) = 0

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

ODE
\[ y''(x) \left (\text {a0}+\text {a1} \sin ^2(y(x))\right )+\text {a2} y(x) \left (\text {a1} \sin ^2(y(x))+\text {a3}\right )+\text {a1} y'(x)^2+\text {a1} y'(x)^2 \sin (y(x)) \cos (y(x))=0 \] ODE Classiﬁcation

[[_2nd_order, _missing_x]]

Book solution method
TO DO

Mathematica ✓
cpu = 9.39979 (sec), leaf count = 384

\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\exp \left (-\text {a1} \left (-\frac {\tan ^{-1}\left (\frac {\sqrt {\text {a0}+\text {a1}} \tan (K[2])}{\sqrt {\text {a0}}}\right )}{\sqrt {\text {a0}} \sqrt {\text {a0}+\text {a1}}}-\frac {\log (2 \text {a0}+\text {a1}-\text {a1} \cos (2 K[2]))}{2 \text {a1}}\right )\right )}{\sqrt {c_1+2 \int _1^{K[2]}\frac {\exp \left (-2 \text {a1} \left (-\frac {\tan ^{-1}\left (\frac {\sqrt {\text {a0}+\text {a1}} \tan (K[1])}{\sqrt {\text {a0}}}\right )}{\sqrt {\text {a0}} \sqrt {\text {a0}+\text {a1}}}-\frac {\log (2 \text {a0}+\text {a1}-\text {a1} \cos (2 K[1]))}{2 \text {a1}}\right )\right ) (-\text {a1} \text {a2} K[1]-2 \text {a2} \text {a3} K[1]+\text {a1} \text {a2} \cos (2 K[1]) K[1])}{2 \text {a0}+\text {a1}-\text {a1} \cos (2 K[1])}dK[1]}}dK[2]\& \right ][x+c_2]\right \},\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\exp \left (-\text {a1} \left (-\frac {\tan ^{-1}\left (\frac {\sqrt {\text {a0}+\text {a1}} \tan (K[3])}{\sqrt {\text {a0}}}\right )}{\sqrt {\text {a0}} \sqrt {\text {a0}+\text {a1}}}-\frac {\log (2 \text {a0}+\text {a1}-\text {a1} \cos (2 K[3]))}{2 \text {a1}}\right )\right )}{\sqrt {c_1+2 \int _1^{K[3]}\frac {\exp \left (-2 \text {a1} \left (-\frac {\tan ^{-1}\left (\frac {\sqrt {\text {a0}+\text {a1}} \tan (K[1])}{\sqrt {\text {a0}}}\right )}{\sqrt {\text {a0}} \sqrt {\text {a0}+\text {a1}}}-\frac {\log (2 \text {a0}+\text {a1}-\text {a1} \cos (2 K[1]))}{2 \text {a1}}\right )\right ) (-\text {a1} \text {a2} K[1]-2 \text {a2} \text {a3} K[1]+\text {a1} \text {a2} \cos (2 K[1]) K[1])}{2 \text {a0}+\text {a1}-\text {a1} \cos (2 K[1])}dK[1]}}dK[3]\& \right ][x+c_2]\right \}\right \}\]

Maple ✓
cpu = 20.898 (sec), leaf count = 1698

\[\text {Expression too large to display}\] 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

{{y[x] -> InverseFunction[Inactive[Integrate][-(1/(E^(a1*(-(ArcTan[(Sqrt[a0 + a1
]*Tan[K[2]])/Sqrt[a0]]/(Sqrt[a0]*Sqrt[a0 + a1])) - Log[2*a0 + a1 - a1*Cos[2*K[2]
]]/(2*a1)))*Sqrt[C[1] + 2*Inactive[Integrate][(-(a1*a2*K[1]) - 2*a2*a3*K[1] + a1
*a2*Cos[2*K[1]]*K[1])/(E^(2*a1*(-(ArcTan[(Sqrt[a0 + a1]*Tan[K[1]])/Sqrt[a0]]/(Sq
rt[a0]*Sqrt[a0 + a1])) - Log[2*a0 + a1 - a1*Cos[2*K[1]]]/(2*a1)))*(2*a0 + a1 - a
1*Cos[2*K[1]])), {K[1], 1, K[2]}]])), {K[2], 1, #1}] & ][x + C[2]]}, {y[x] -> In
verseFunction[Inactive[Integrate][1/(E^(a1*(-(ArcTan[(Sqrt[a0 + a1]*Tan[K[3]])/S
qrt[a0]]/(Sqrt[a0]*Sqrt[a0 + a1])) - Log[2*a0 + a1 - a1*Cos[2*K[3]]]/(2*a1)))*Sq
rt[C[1] + 2*Inactive[Integrate][(-(a1*a2*K[1]) - 2*a2*a3*K[1] + a1*a2*Cos[2*K[1]
]*K[1])/(E^(2*a1*(-(ArcTan[(Sqrt[a0 + a1]*Tan[K[1]])/Sqrt[a0]]/(Sqrt[a0]*Sqrt[a0
+ a1])) - Log[2*a0 + a1 - a1*Cos[2*K[1]]]/(2*a1)))*(2*a0 + a1 - a1*Cos[2*K[1]])
), {K[1], 1, K[3]}]]), {K[3], 1, #1}] & ][x + C[2]]}}

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))

Maple raw output

[Intat((tan(_h)^2*a0+a1*tan(_h)^2+a0)/(-(tan(_h)^2*a0+a1*tan(_h)^2+a0)*(2*tan(_h
)^2*a0*a1*a2*Int(1/(1+tan(_h)^2)*exp(a1/(a0^2+a1*a0)^(1/2)*arctan(tan(_h)*(a0+a1
)/(a0^2+a1*a0)^(1/2)))^2*tan(_h)^2*_h*sin(_h)^2/(a0+a1*sin(_h)^2),_h)+2*tan(_h)^
2*a1^2*a2*Int(1/(1+tan(_h)^2)*exp(a1/(a0^2+a1*a0)^(1/2)*arctan(tan(_h)*(a0+a1)/(
a0^2+a1*a0)^(1/2)))^2*tan(_h)^2*_h*sin(_h)^2/(a0+a1*sin(_h)^2),_h)+2*tan(_h)^2*a
0*a1*a2*Int(1/(1+tan(_h)^2)*exp(a1/(a0^2+a1*a0)^(1/2)*arctan(tan(_h)*(a0+a1)/(a0
^2+a1*a0)^(1/2)))^2*_h*sin(_h)^2/(a0+a1*sin(_h)^2),_h)+2*tan(_h)^2*a0*a2*a3*Int(
1/(1+tan(_h)^2)*exp(a1/(a0^2+a1*a0)^(1/2)*arctan(tan(_h)*(a0+a1)/(a0^2+a1*a0)^(1
/2)))^2*tan(_h)^2*_h/(a0+a1*sin(_h)^2),_h)+2*tan(_h)^2*a1*a2*a3*Int(1/(1+tan(_h)
^2)*exp(a1/(a0^2+a1*a0)^(1/2)*arctan(tan(_h)*(a0+a1)/(a0^2+a1*a0)^(1/2)))^2*tan(
_h)^2*_h/(a0+a1*sin(_h)^2),_h)+2*tan(_h)^2*a0*a2*a3*Int(1/(1+tan(_h)^2)*exp(a1/(
a0^2+a1*a0)^(1/2)*arctan(tan(_h)*(a0+a1)/(a0^2+a1*a0)^(1/2)))^2*_h/(a0+a1*sin(_h
)^2),_h)+2*a1*a0*a2*Int(1/(1+tan(_h)^2)*exp(a1/(a0^2+a1*a0)^(1/2)*arctan(tan(_h)
*(a0+a1)/(a0^2+a1*a0)^(1/2)))^2*tan(_h)^2*_h*sin(_h)^2/(a0+a1*sin(_h)^2),_h)+2*a
1^2*a2*Int(1/(1+tan(_h)^2)*exp(a1/(a0^2+a1*a0)^(1/2)*arctan(tan(_h)*(a0+a1)/(a0^
2+a1*a0)^(1/2)))^2*tan(_h)^2*_h*sin(_h)^2/(a0+a1*sin(_h)^2),_h)+2*a1*a0*a2*Int(1
/(1+tan(_h)^2)*exp(a1/(a0^2+a1*a0)^(1/2)*arctan(tan(_h)*(a0+a1)/(a0^2+a1*a0)^(1/
2)))^2*_h*sin(_h)^2/(a0+a1*sin(_h)^2),_h)+2*a0*a2*a3*Int(1/(1+tan(_h)^2)*exp(a1/
(a0^2+a1*a0)^(1/2)*arctan(tan(_h)*(a0+a1)/(a0^2+a1*a0)^(1/2)))^2*tan(_h)^2*_h/(a
0+a1*sin(_h)^2),_h)+2*a2*a3*a1*Int(1/(1+tan(_h)^2)*exp(a1/(a0^2+a1*a0)^(1/2)*arc
tan(tan(_h)*(a0+a1)/(a0^2+a1*a0)^(1/2)))^2*tan(_h)^2*_h/(a0+a1*sin(_h)^2),_h)+2*
a0*a2*a3*Int(1/(1+tan(_h)^2)*exp(a1/(a0^2+a1*a0)^(1/2)*arctan(tan(_h)*(a0+a1)/(a
0^2+a1*a0)^(1/2)))^2*_h/(a0+a1*sin(_h)^2),_h)-tan(_h)^2*_C1-_C1))^(1/2)*exp(a1/(
a0*(a0+a1))^(1/2)*arctan(tan(_h)*(a0+a1)/(a0*(a0+a1))^(1/2))),_h = y(x))-x-_C2 =
0, Intat(-(tan(_h)^2*a0+a1*tan(_h)^2+a0)/(-(tan(_h)^2*a0+a1*tan(_h)^2+a0)*(2*ta
n(_h)^2*a0*a1*a2*Int(1/(1+tan(_h)^2)*exp(a1/(a0^2+a1*a0)^(1/2)*arctan(tan(_h)*(a
0+a1)/(a0^2+a1*a0)^(1/2)))^2*tan(_h)^2*_h*sin(_h)^2/(a0+a1*sin(_h)^2),_h)+2*tan(
_h)^2*a1^2*a2*Int(1/(1+tan(_h)^2)*exp(a1/(a0^2+a1*a0)^(1/2)*arctan(tan(_h)*(a0+a
1)/(a0^2+a1*a0)^(1/2)))^2*tan(_h)^2*_h*sin(_h)^2/(a0+a1*sin(_h)^2),_h)+2*tan(_h)
^2*a0*a1*a2*Int(1/(1+tan(_h)^2)*exp(a1/(a0^2+a1*a0)^(1/2)*arctan(tan(_h)*(a0+a1)
/(a0^2+a1*a0)^(1/2)))^2*_h*sin(_h)^2/(a0+a1*sin(_h)^2),_h)+2*tan(_h)^2*a0*a2*a3*
Int(1/(1+tan(_h)^2)*exp(a1/(a0^2+a1*a0)^(1/2)*arctan(tan(_h)*(a0+a1)/(a0^2+a1*a0
)^(1/2)))^2*tan(_h)^2*_h/(a0+a1*sin(_h)^2),_h)+2*tan(_h)^2*a1*a2*a3*Int(1/(1+tan
(_h)^2)*exp(a1/(a0^2+a1*a0)^(1/2)*arctan(tan(_h)*(a0+a1)/(a0^2+a1*a0)^(1/2)))^2*
tan(_h)^2*_h/(a0+a1*sin(_h)^2),_h)+2*tan(_h)^2*a0*a2*a3*Int(1/(1+tan(_h)^2)*exp(
a1/(a0^2+a1*a0)^(1/2)*arctan(tan(_h)*(a0+a1)/(a0^2+a1*a0)^(1/2)))^2*_h/(a0+a1*si
n(_h)^2),_h)+2*a1*a0*a2*Int(1/(1+tan(_h)^2)*exp(a1/(a0^2+a1*a0)^(1/2)*arctan(tan
(_h)*(a0+a1)/(a0^2+a1*a0)^(1/2)))^2*tan(_h)^2*_h*sin(_h)^2/(a0+a1*sin(_h)^2),_h)
+2*a1^2*a2*Int(1/(1+tan(_h)^2)*exp(a1/(a0^2+a1*a0)^(1/2)*arctan(tan(_h)*(a0+a1)/
(a0^2+a1*a0)^(1/2)))^2*tan(_h)^2*_h*sin(_h)^2/(a0+a1*sin(_h)^2),_h)+2*a1*a0*a2*I
nt(1/(1+tan(_h)^2)*exp(a1/(a0^2+a1*a0)^(1/2)*arctan(tan(_h)*(a0+a1)/(a0^2+a1*a0)
^(1/2)))^2*_h*sin(_h)^2/(a0+a1*sin(_h)^2),_h)+2*a0*a2*a3*Int(1/(1+tan(_h)^2)*exp
(a1/(a0^2+a1*a0)^(1/2)*arctan(tan(_h)*(a0+a1)/(a0^2+a1*a0)^(1/2)))^2*tan(_h)^2*_
h/(a0+a1*sin(_h)^2),_h)+2*a2*a3*a1*Int(1/(1+tan(_h)^2)*exp(a1/(a0^2+a1*a0)^(1/2)
*arctan(tan(_h)*(a0+a1)/(a0^2+a1*a0)^(1/2)))^2*tan(_h)^2*_h/(a0+a1*sin(_h)^2),_h
)+2*a0*a2*a3*Int(1/(1+tan(_h)^2)*exp(a1/(a0^2+a1*a0)^(1/2)*arctan(tan(_h)*(a0+a1
)/(a0^2+a1*a0)^(1/2)))^2*_h/(a0+a1*sin(_h)^2),_h)-tan(_h)^2*_C1-_C1))^(1/2)*exp(
a1/(a0*(a0+a1))^(1/2)*arctan(tan(_h)*(a0+a1)/(a0*(a0+a1))^(1/2))),_h = y(x))-x-_
C2 = 0]

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4.41.45 \(y''(x) \left (\text {a0}+\text {a1} \sin ^2(y(x))\right )+\text {a2} y(x) \left (\text {a1} \sin ^2(y(x))+\text {a3}\right )+\text {a1} y'(x)^2+\text {a1} y'(x)^2 \sin (y(x)) \cos (y(x))=0\)