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 = 600. (sec), leaf count = 0 , timed out
$Aborted
Maple ✓
cpu = 3.671 (sec), leaf count = 818
\[ \left \{ \int ^{y \left ( x \right ) }\!{( \left ( {\it a0}+{\it a1} \right ) \left ( \tan \left ( {\it \_h} \right ) \right ) ^{2}+{\it a0}){{\rm e}^{{{\it a1}\arctan \left ( { \left ( {\it a0}+{\it a1} \right ) \tan \left ( {\it \_h} \right ) {\frac {1}{\sqrt {{\it a0}\, \left ( {\it a0}+{\it a1} \right ) }}}} \right ) {\frac {1}{\sqrt {{\it a0}\, \left ( {\it a0}+{\it a1} \right ) }}}}}}{\frac {1}{\sqrt {-2\, \left ( \left ( \left ( \int \!{\frac { \left ( \tan \left ( {\it \_h} \right ) \right ) ^{2} \left ( \sin \left ( {\it \_h} \right ) \right ) ^{2}{\it \_h}}{ \left ( \left ( \tan \left ( {\it \_h} \right ) \right ) ^{2}+1 \right ) \left ( {\it a0}+{\it a1}\, \left ( \sin \left ( {\it \_h} \right ) \right ) ^{2} \right ) } \left ( {{\rm e}^{{\frac {{\it a1}}{\sqrt {{\it a0}\, \left ( {\it a0}+{\it a1} \right ) }}\arctan \left ( {\frac { \left ( {\it a0}+{\it a1} \right ) \tan \left ( {\it \_h} \right ) }{\sqrt {{\it a0}\, \left ( {\it a0}+{\it a1} \right ) }}} \right ) }}} \right ) ^{2}}\,{\rm d}{\it \_h}+\int \!{\frac { \left ( \sin \left ( {\it \_h} \right ) \right ) ^{2}{\it \_h}}{ \left ( \left ( \tan \left ( {\it \_h} \right ) \right ) ^{2}+1 \right ) \left ( {\it a0}+{\it a1}\, \left ( \sin \left ( {\it \_h} \right ) \right ) ^{2} \right ) } \left ( {{\rm e}^{{\frac {{\it a1}}{\sqrt {{\it a0}\, \left ( {\it a0}+{\it a1} \right ) }}\arctan \left ( {\frac { \left ( {\it a0}+{\it a1} \right ) \tan \left ( {\it \_h} \right ) }{\sqrt {{\it a0}\, \left ( {\it a0}+{\it a1} \right ) }}} \right ) }}} \right ) ^{2}}\,{\rm d}{\it \_h} \right ) {\it a1}+ \left ( \int \!{\frac { \left ( \tan \left ( {\it \_h} \right ) \right ) ^{2}{\it \_h}}{ \left ( \left ( \tan \left ( {\it \_h} \right ) \right ) ^{2}+1 \right ) \left ( {\it a0}+{\it a1}\, \left ( \sin \left ( {\it \_h} \right ) \right ) ^{2} \right ) } \left ( {{\rm e}^{{\frac {{\it a1}}{\sqrt {{\it a0}\, \left ( {\it a0}+{\it a1} \right ) }}\arctan \left ( {\frac { \left ( {\it a0}+{\it a1} \right ) \tan \left ( {\it \_h} \right ) }{\sqrt {{\it a0}\, \left ( {\it a0}+{\it a1} \right ) }}} \right ) }}} \right ) ^{2}}\,{\rm d}{\it \_h}+\int \!{\frac {{\it \_h}}{ \left ( \left ( \tan \left ( {\it \_h} \right ) \right ) ^{2}+1 \right ) \left ( {\it a0}+{\it a1}\, \left ( \sin \left ( {\it \_h} \right ) \right ) ^{2} \right ) } \left ( {{\rm e}^{{\frac {{\it a1}}{\sqrt {{\it a0}\, \left ( {\it a0}+{\it a1} \right ) }}\arctan \left ( {\frac { \left ( {\it a0}+{\it a1} \right ) \tan \left ( {\it \_h} \right ) }{\sqrt {{\it a0}\, \left ( {\it a0}+{\it a1} \right ) }}} \right ) }}} \right ) ^{2}}\,{\rm d}{\it \_h} \right ) {\it a3} \right ) {\it a2}\,{\it a0}+{{\it a1}}^{2}{\it a2}\,\int \!{\frac { \left ( \tan \left ( {\it \_h} \right ) \right ) ^{2} \left ( \sin \left ( {\it \_h} \right ) \right ) ^{2}{\it \_h}}{ \left ( \left ( \tan \left ( {\it \_h} \right ) \right ) ^{2}+1 \right ) \left ( {\it a0}+{\it a1}\, \left ( \sin \left ( {\it \_h} \right ) \right ) ^{2} \right ) } \left ( {{\rm e}^{{\frac {{\it a1}}{\sqrt {{\it a0}\, \left ( {\it a0}+{\it a1} \right ) }}\arctan \left ( {\frac { \left ( {\it a0}+{\it a1} \right ) \tan \left ( {\it \_h} \right ) }{\sqrt {{\it a0}\, \left ( {\it a0}+{\it a1} \right ) }}} \right ) }}} \right ) ^{2}}\,{\rm d}{\it \_h}+{\it a1}\,{\it a2}\,{\it a3}\,\int \!{\frac { \left ( \tan \left ( {\it \_h} \right ) \right ) ^{2}{\it \_h}}{ \left ( \left ( \tan \left ( {\it \_h} \right ) \right ) ^{2}+1 \right ) \left ( {\it a0}+{\it a1}\, \left ( \sin \left ( {\it \_h} \right ) \right ) ^{2} \right ) } \left ( {{\rm e}^{{\frac {{\it a1}}{\sqrt {{\it a0}\, \left ( {\it a0}+{\it a1} \right ) }}\arctan \left ( {\frac { \left ( {\it a0}+{\it a1} \right ) \tan \left ( {\it \_h} \right ) }{\sqrt {{\it a0}\, \left ( {\it a0}+{\it a1} \right ) }}} \right ) }}} \right ) ^{2}}\,{\rm d}{\it \_h}-{\it \_C1}/2 \right ) \left ( \left ( \tan \left ( {\it \_h} \right ) \right ) ^{2}+1 \right ) \left ( \left ( {\it a0}+{\it a1} \right ) \left ( \tan \left ( {\it \_h} \right ) \right ) ^{2}+{\it a0} \right ) }}}}{d{\it \_h}}-x-{\it \_C2}=0,\int ^{y \left ( x \right ) }\!-{( \left ( {\it a0}+{\it a1} \right ) \left ( \tan \left ( {\it \_h} \right ) \right ) ^{2}+{\it a0}){{\rm e}^{{{\it a1}\arctan \left ( { \left ( {\it a0}+{\it a1} \right ) \tan \left ( {\it \_h} \right ) {\frac {1}{\sqrt {{\it a0}\, \left ( {\it a0}+{\it a1} \right ) }}}} \right ) {\frac {1}{\sqrt {{\it a0}\, \left ( {\it a0}+{\it a1} \right ) }}}}}}{\frac {1}{\sqrt {-2\, \left ( \left ( \left ( \int \!{\frac { \left ( \tan \left ( {\it \_h} \right ) \right ) ^{2} \left ( \sin \left ( {\it \_h} \right ) \right ) ^{2}{\it \_h}}{ \left ( \left ( \tan \left ( {\it \_h} \right ) \right ) ^{2}+1 \right ) \left ( {\it a0}+{\it a1}\, \left ( \sin \left ( {\it \_h} \right ) \right ) ^{2} \right ) } \left ( {{\rm e}^{{\frac {{\it a1}}{\sqrt {{\it a0}\, \left ( {\it a0}+{\it a1} \right ) }}\arctan \left ( {\frac { \left ( {\it a0}+{\it a1} \right ) \tan \left ( {\it \_h} \right ) }{\sqrt {{\it a0}\, \left ( {\it a0}+{\it a1} \right ) }}} \right ) }}} \right ) ^{2}}\,{\rm d}{\it \_h}+\int \!{\frac { \left ( \sin \left ( {\it \_h} \right ) \right ) ^{2}{\it \_h}}{ \left ( \left ( \tan \left ( {\it \_h} \right ) \right ) ^{2}+1 \right ) \left ( {\it a0}+{\it a1}\, \left ( \sin \left ( {\it \_h} \right ) \right ) ^{2} \right ) } \left ( {{\rm e}^{{\frac {{\it a1}}{\sqrt {{\it a0}\, \left ( {\it a0}+{\it a1} \right ) }}\arctan \left ( {\frac { \left ( {\it a0}+{\it a1} \right ) \tan \left ( {\it \_h} \right ) }{\sqrt {{\it a0}\, \left ( {\it a0}+{\it a1} \right ) }}} \right ) }}} \right ) ^{2}}\,{\rm d}{\it \_h} \right ) {\it a1}+ \left ( \int \!{\frac { \left ( \tan \left ( {\it \_h} \right ) \right ) ^{2}{\it \_h}}{ \left ( \left ( \tan \left ( {\it \_h} \right ) \right ) ^{2}+1 \right ) \left ( {\it a0}+{\it a1}\, \left ( \sin \left ( {\it \_h} \right ) \right ) ^{2} \right ) } \left ( {{\rm e}^{{\frac {{\it a1}}{\sqrt {{\it a0}\, \left ( {\it a0}+{\it a1} \right ) }}\arctan \left ( {\frac { \left ( {\it a0}+{\it a1} \right ) \tan \left ( {\it \_h} \right ) }{\sqrt {{\it a0}\, \left ( {\it a0}+{\it a1} \right ) }}} \right ) }}} \right ) ^{2}}\,{\rm d}{\it \_h}+\int \!{\frac {{\it \_h}}{ \left ( \left ( \tan \left ( {\it \_h} \right ) \right ) ^{2}+1 \right ) \left ( {\it a0}+{\it a1}\, \left ( \sin \left ( {\it \_h} \right ) \right ) ^{2} \right ) } \left ( {{\rm e}^{{\frac {{\it a1}}{\sqrt {{\it a0}\, \left ( {\it a0}+{\it a1} \right ) }}\arctan \left ( {\frac { \left ( {\it a0}+{\it a1} \right ) \tan \left ( {\it \_h} \right ) }{\sqrt {{\it a0}\, \left ( {\it a0}+{\it a1} \right ) }}} \right ) }}} \right ) ^{2}}\,{\rm d}{\it \_h} \right ) {\it a3} \right ) {\it a2}\,{\it a0}+{{\it a1}}^{2}{\it a2}\,\int \!{\frac { \left ( \tan \left ( {\it \_h} \right ) \right ) ^{2} \left ( \sin \left ( {\it \_h} \right ) \right ) ^{2}{\it \_h}}{ \left ( \left ( \tan \left ( {\it \_h} \right ) \right ) ^{2}+1 \right ) \left ( {\it a0}+{\it a1}\, \left ( \sin \left ( {\it \_h} \right ) \right ) ^{2} \right ) } \left ( {{\rm e}^{{\frac {{\it a1}}{\sqrt {{\it a0}\, \left ( {\it a0}+{\it a1} \right ) }}\arctan \left ( {\frac { \left ( {\it a0}+{\it a1} \right ) \tan \left ( {\it \_h} \right ) }{\sqrt {{\it a0}\, \left ( {\it a0}+{\it a1} \right ) }}} \right ) }}} \right ) ^{2}}\,{\rm d}{\it \_h}+{\it a1}\,{\it a2}\,{\it a3}\,\int \!{\frac { \left ( \tan \left ( {\it \_h} \right ) \right ) ^{2}{\it \_h}}{ \left ( \left ( \tan \left ( {\it \_h} \right ) \right ) ^{2}+1 \right ) \left ( {\it a0}+{\it a1}\, \left ( \sin \left ( {\it \_h} \right ) \right ) ^{2} \right ) } \left ( {{\rm e}^{{\frac {{\it a1}}{\sqrt {{\it a0}\, \left ( {\it a0}+{\it a1} \right ) }}\arctan \left ( {\frac { \left ( {\it a0}+{\it a1} \right ) \tan \left ( {\it \_h} \right ) }{\sqrt {{\it a0}\, \left ( {\it a0}+{\it a1} \right ) }}} \right ) }}} \right ) ^{2}}\,{\rm d}{\it \_h}-{\it \_C1}/2 \right ) \left ( \left ( \tan \left ( {\it \_h} \right ) \right ) ^{2}+1 \right ) \left ( \left ( {\it a0}+{\it a1} \right ) \left ( \tan \left ( {\it \_h} \right ) \right ) ^{2}+{\it a0} \right ) }}}}{d{\it \_h}}-x-{\it \_C2}=0 \right \} \] 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*sin(_h)^2)*sin(_h)^2*_h,_h))*a1+(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)+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*a2*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)+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)*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*sin(_h)^2)*sin(_h)^2*_h,_h))*a1+(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)+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*a2*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)+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