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 = 0.934241 (sec), leaf count = 375
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [-\frac {4 \sqrt [4]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{3/4} \, _2F_1\left (\frac {1}{4},\frac {3}{4};\frac {5}{4};\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$1})^{3/4}}\& \right ]\left [\int _1^x-\sqrt [4]{-1} \sqrt [4]{f(K[1])}dK[1]+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [-\frac {4 \sqrt [4]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{3/4} \, _2F_1\left (\frac {1}{4},\frac {3}{4};\frac {5}{4};\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$1})^{3/4}}\& \right ]\left [\int _1^x\sqrt [4]{-1} \sqrt [4]{f(K[2])}dK[2]+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [-\frac {4 \sqrt [4]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{3/4} \, _2F_1\left (\frac {1}{4},\frac {3}{4};\frac {5}{4};\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$1})^{3/4}}\& \right ]\left [\int _1^x-(-1)^{3/4} \sqrt [4]{f(K[3])}dK[3]+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [-\frac {4 \sqrt [4]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{3/4} \, _2F_1\left (\frac {1}{4},\frac {3}{4};\frac {5}{4};\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$1})^{3/4}}\& \right ]\left [\int _1^x(-1)^{3/4} \sqrt [4]{f(K[4])}dK[4]+c_1\right ]\right \}\right \}\]
Maple ✓
cpu = 2.819 (sec), leaf count = 262
\[\left [\int _{}^{y \left (x \right )}\frac {1}{\left (\left (-\textit {\_a} +a \right ) \left (-\textit {\_a} +b \right )\right )^{\frac {3}{4}}}d \textit {\_a} +\int _{}^{x}-\frac {\left (-f \left (\textit {\_a} \right ) \left (b -y \left (x \right )\right )^{3} \left (a -y \left (x \right )\right )^{3}\right )^{\frac {1}{4}}}{\left (\left (a -y \left (x \right )\right ) \left (b -y \left (x \right )\right )\right )^{\frac {3}{4}}}d \textit {\_a} +\textit {\_C1} = 0, \int _{}^{y \left (x \right )}\frac {1}{\left (\left (-\textit {\_a} +a \right ) \left (-\textit {\_a} +b \right )\right )^{\frac {3}{4}}}d \textit {\_a} +\int _{}^{x}\frac {i \left (-f \left (\textit {\_a} \right ) \left (b -y \left (x \right )\right )^{3} \left (a -y \left (x \right )\right )^{3}\right )^{\frac {1}{4}}}{\left (\left (a -y \left (x \right )\right ) \left (b -y \left (x \right )\right )\right )^{\frac {3}{4}}}d \textit {\_a} +\textit {\_C1} = 0, \int _{}^{y \left (x \right )}\frac {1}{\left (\left (-\textit {\_a} +a \right ) \left (-\textit {\_a} +b \right )\right )^{\frac {3}{4}}}d \textit {\_a} +\int _{}^{x}-\frac {i \left (-f \left (\textit {\_a} \right ) \left (b -y \left (x \right )\right )^{3} \left (a -y \left (x \right )\right )^{3}\right )^{\frac {1}{4}}}{\left (\left (a -y \left (x \right )\right ) \left (b -y \left (x \right )\right )\right )^{\frac {3}{4}}}d \textit {\_a} +\textit {\_C1} = 0, \int _{}^{y \left (x \right )}\frac {1}{\left (\left (-\textit {\_a} +a \right ) \left (-\textit {\_a} +b \right )\right )^{\frac {3}{4}}}d \textit {\_a} +\int _{}^{x}\frac {\left (-f \left (\textit {\_a} \right ) \left (b -y \left (x \right )\right )^{3} \left (a -y \left (x \right )\right )^{3}\right )^{\frac {1}{4}}}{\left (\left (a -y \left (x \right )\right ) \left (b -y \left (x \right )\right )\right )^{\frac {3}{4}}}d \textit {\_a} +\textit {\_C1} = 0\right ]\] 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] + Inactive[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] + Inactive[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] + Inactive[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] + Inactive[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))
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]