ODE
\[ f(x) (y(x)-a)^3 (y(x)-b)^3 (y(x)-c)^2+y'(x)^4=0 \] 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 = 1.59618 (sec), leaf count = 547
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\frac {4 \sqrt [4]{a-\text {$\#$1}} \sqrt {c-\text {$\#$1}} \left (\frac {(b-\text {$\#$1}) (a-c)}{(c-\text {$\#$1}) (a-b)}\right )^{3/4} \, _2F_1\left (\frac {1}{4},\frac {3}{4};\frac {5}{4};\frac {(c-b) (a-\text {$\#$1})}{(a-b) (c-\text {$\#$1})}\right )}{(b-\text {$\#$1})^{3/4} (a-c)}\& \right ]\left [\int _1^x-\sqrt [4]{-1} \sqrt [4]{f(K[1])}dK[1]+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {4 \sqrt [4]{a-\text {$\#$1}} \sqrt {c-\text {$\#$1}} \left (\frac {(b-\text {$\#$1}) (a-c)}{(c-\text {$\#$1}) (a-b)}\right )^{3/4} \, _2F_1\left (\frac {1}{4},\frac {3}{4};\frac {5}{4};\frac {(c-b) (a-\text {$\#$1})}{(a-b) (c-\text {$\#$1})}\right )}{(b-\text {$\#$1})^{3/4} (a-c)}\& \right ]\left [\int _1^x\sqrt [4]{-1} \sqrt [4]{f(K[2])}dK[2]+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {4 \sqrt [4]{a-\text {$\#$1}} \sqrt {c-\text {$\#$1}} \left (\frac {(b-\text {$\#$1}) (a-c)}{(c-\text {$\#$1}) (a-b)}\right )^{3/4} \, _2F_1\left (\frac {1}{4},\frac {3}{4};\frac {5}{4};\frac {(c-b) (a-\text {$\#$1})}{(a-b) (c-\text {$\#$1})}\right )}{(b-\text {$\#$1})^{3/4} (a-c)}\& \right ]\left [\int _1^x-(-1)^{3/4} \sqrt [4]{f(K[3])}dK[3]+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {4 \sqrt [4]{a-\text {$\#$1}} \sqrt {c-\text {$\#$1}} \left (\frac {(b-\text {$\#$1}) (a-c)}{(c-\text {$\#$1}) (a-b)}\right )^{3/4} \, _2F_1\left (\frac {1}{4},\frac {3}{4};\frac {5}{4};\frac {(c-b) (a-\text {$\#$1})}{(a-b) (c-\text {$\#$1})}\right )}{(b-\text {$\#$1})^{3/4} (a-c)}\& \right ]\left [\int _1^x(-1)^{3/4} \sqrt [4]{f(K[4])}dK[4]+c_1\right ]\right \}\right \}\]
Maple ✓
cpu = 1.223 (sec), leaf count = 92
\[\left [\int _{}^{y \left (x \right )}\frac {1}{\sqrt {\textit {\_a} -c}\, \left (\textit {\_a} -a \right )^{\frac {3}{4}} \left (\textit {\_a} -b \right )^{\frac {3}{4}}}d \textit {\_a} +\int _{}^{x}-\frac {\left (-f \left (\textit {\_a} \right ) \left (c -y \left (x \right )\right )^{2} \left (b -y \left (x \right )\right )^{3} \left (a -y \left (x \right )\right )^{3}\right )^{\frac {1}{4}}}{\sqrt {y \left (x \right )-c}\, \left (y \left (x \right )-a \right )^{\frac {3}{4}} \left (y \left (x \right )-b \right )^{\frac {3}{4}}}d \textit {\_a} +\textit {\_C1} = 0\right ]\] Mathematica raw input
DSolve[f[x]*(-a + y[x])^3*(-b + y[x])^3*(-c + y[x])^2 + y'[x]^4 == 0,y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[(4*Hypergeometric2F1[1/4, 3/4, 5/4, ((-b + c)*(a - #1))/((a - b)*(c - #1))]*(a - #1)^(1/4)*(((a - c)*(b - #1))/((a - b)*(c - #1)))^(3/4)*Sqrt[c - #1])/((a - c)*(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, ((-b + c)*(a - #1))/((a - b)*(c - #1))]*(a - #1)^(1/4)*(((a - c)*(b - #1))/((a - b)*(c - #1)))^(3/4)*Sqrt[c - #1])/((a - c)*(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, ((-b + c)*(a - #1))/((a - b)*(c - #1))]*(a - #1)^(1/4)*(((a - c)*(b - #1))/((a - b)*(c - #1)))^(3/4)*Sqrt[c - #1])/((a - c)*(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, ((-b + c)*(a - #1))/((a - b)*(c - #1))]*(a - #1)^(1/4)*(((a - c)*(b - #1))/((a - b)*(c - #1)))^(3/4)*Sqrt[c - #1])/((a - c)*(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*(y(x)-c)^2 = 0, y(x))
Maple raw output
[Intat(1/(_a-c)^(1/2)/(_a-a)^(3/4)/(_a-b)^(3/4),_a = y(x))+Intat(-(-f(_a)*(c-y(x))^2*(b-y(x))^3*(a-y(x))^3)^(1/4)/(y(x)-c)^(1/2)/(y(x)-a)^(3/4)/(y(x)-b)^(3/4),_a = x)+_C1 = 0]