ODE
\[ f(x) (y(x)-a)^3 (y(x)-b)^3+y'(x)^4=0 \] ODE Classification
[[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
Book solution method
Binomial equation \((y')^m + F(x) G(y)=0\)
Mathematica ✓
cpu = 1.27277 (sec), leaf count = 371
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [-\frac {4 \sqrt [4]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{3/4} \, _2F_1\left (\frac {1}{4},\frac {3}{4};\frac {5}{4};\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$1})^{3/4}}\& \right ]\left [\int _1^x -\sqrt [4]{-1} \sqrt [4]{f(K[1])} \, dK[1]+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [-\frac {4 \sqrt [4]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{3/4} \, _2F_1\left (\frac {1}{4},\frac {3}{4};\frac {5}{4};\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$1})^{3/4}}\& \right ]\left [\int _1^x \sqrt [4]{-1} \sqrt [4]{f(K[2])} \, dK[2]+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [-\frac {4 \sqrt [4]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{3/4} \, _2F_1\left (\frac {1}{4},\frac {3}{4};\frac {5}{4};\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$1})^{3/4}}\& \right ]\left [\int _1^x -(-1)^{3/4} \sqrt [4]{f(K[3])} \, dK[3]+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [-\frac {4 \sqrt [4]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{3/4} \, _2F_1\left (\frac {1}{4},\frac {3}{4};\frac {5}{4};\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$1})^{3/4}}\& \right ]\left [\int _1^x (-1)^{3/4} \sqrt [4]{f(K[4])} \, dK[4]+c_1\right ]\right \}\right \}\]
Maple ✓
cpu = 0.443 (sec), leaf count = 262
\[ \left \{ \int ^{y \left ( x \right ) }\! \left ( \left ( -{\it \_a}+a \right ) \left ( -{\it \_a}+b \right ) \right ) ^{-{\frac {3}{4}}}{d{\it \_a}}+\int ^{x}\!{1\sqrt [4]{-f \left ( {\it \_a} \right ) \left ( b-y \left ( x \right ) \right ) ^{3} \left ( a-y \left ( x \right ) \right ) ^{3}} \left ( \left ( a-y \left ( x \right ) \right ) \left ( b-y \left ( x \right ) \right ) \right ) ^{-{\frac {3}{4}}}}{d{\it \_a}}+{\it \_C1}=0,\int ^{y \left ( x \right ) }\! \left ( \left ( -{\it \_a}+a \right ) \left ( -{\it \_a}+b \right ) \right ) ^{-{\frac {3}{4}}}{d{\it \_a}}+\int ^{x}\!{-i\sqrt [4]{-f \left ( {\it \_a} \right ) \left ( b-y \left ( x \right ) \right ) ^{3} \left ( a-y \left ( x \right ) \right ) ^{3}} \left ( \left ( a-y \left ( x \right ) \right ) \left ( b-y \left ( x \right ) \right ) \right ) ^{-{\frac {3}{4}}}}{d{\it \_a}}+{\it \_C1}=0,\int ^{y \left ( x \right ) }\! \left ( \left ( -{\it \_a}+a \right ) \left ( -{\it \_a}+b \right ) \right ) ^{-{\frac {3}{4}}}{d{\it \_a}}+\int ^{x}\!{i\sqrt [4]{-f \left ( {\it \_a} \right ) \left ( b-y \left ( x \right ) \right ) ^{3} \left ( a-y \left ( x \right ) \right ) ^{3}} \left ( \left ( a-y \left ( x \right ) \right ) \left ( b-y \left ( x \right ) \right ) \right ) ^{-{\frac {3}{4}}}}{d{\it \_a}}+{\it \_C1}=0,\int ^{y \left ( x \right ) }\! \left ( \left ( -{\it \_a}+a \right ) \left ( -{\it \_a}+b \right ) \right ) ^{-{\frac {3}{4}}}{d{\it \_a}}+\int ^{x}\!-{1\sqrt [4]{-f \left ( {\it \_a} \right ) \left ( b-y \left ( x \right ) \right ) ^{3} \left ( a-y \left ( x \right ) \right ) ^{3}} \left ( \left ( a-y \left ( x \right ) \right ) \left ( b-y \left ( x \right ) \right ) \right ) ^{-{\frac {3}{4}}}}{d{\it \_a}}+{\it \_C1}=0 \right \} \] Mathematica raw input
DSolve[f[x]*(-a + y[x])^3*(-b + y[x])^3 + y'[x]^4 == 0,y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[(-4*Hypergeometric2F1[1/4, 3/4, 5/4, (a - #1)/(a - b)]*(a - #1)^(1/4)*((-b + #1)/(a - b))^(3/4))/(b - #1)^(3/4) & ][C[1] + Integrate[-((-1)^(1/4)*f[K[1]]^(1/4)), {K[1], 1, x}]]}, {y[x] -> InverseFunction[(-4*Hypergeometric2F1[1/4, 3/4, 5/4, (a - #1)/(a - b)]*(a - #1)^(1/4)*((-b + #1)/(a - b))^(3/4))/(b - #1)^(3/4) & ][C[1] + Integrate[(-1)^(1/4)*f[K[2]]^(1/4), {K[2], 1, x}]]}, {y[x] -> InverseFunction[(-4*Hypergeometric2F1[1/4, 3/4, 5/4, (a - #1)/(a - b)]*(a - #1)^(1/4)*((-b + #1)/(a - b))^(3/4))/(b - #1)^(3/4) & ][C[1] + Integrate[-((-1)^(3/4)*f[K[3]]^(1/4)), {K[3], 1, x}]]}, {y[x] -> InverseFunction[(-4*Hypergeometric2F1[1/4, 3/4, 5/4, (a - #1)/(a - b)]*(a - #1)^(1/4)*((-b + #1)/(a - b))^(3/4))/(b - #1)^(3/4) & ][C[1] + Integrate[(-1)^(3/4)*f[K[4]]^(1/4), {K[4], 1, x}]]}}
Maple raw input
dsolve(diff(y(x),x)^4+f(x)*(y(x)-a)^3*(y(x)-b)^3 = 0, y(x),'implicit')
Maple raw output
Intat(1/((-_a+a)*(-_a+b))^(3/4),_a = y(x))+Intat(-(-f(_a)*(b-y(x))^3*(a-y(x))^3)^(1/4)/((a-y(x))*(b-y(x)))^(3/4),_a = x)+_C1 = 0, Intat(1/((-_a+a)*(-_a+b))^(3/4),_a = y(x))+Intat(I*(-f(_a)*(b-y(x))^3*(a-y(x))^3)^(1/4)/((a-y(x))*(b-y(x)))^(3/4),_a = x)+_C1 = 0, Intat(1/((-_a+a)*(-_a+b))^(3/4),_a = y(x))+Intat(-I*(-f(_a)*(b-y(x))^3*(a-y(x))^3)^(1/4)/((a-y(x))*(b-y(x)))^(3/4),_a = x)+_C1 = 0, Intat(1/((-_a+a)*(-_a+b))^(3/4),_a = y(x))+Intat((-f(_a)*(b-y(x))^3*(a-y(x))^3)^(1/4)/((a-y(x))*(b-y(x)))^(3/4),_a = x)+_C1 = 0