ODE
\[ f(x) (y(x)-a)^5 (y(x)-b)^4+y'(x)^6=0 \] 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 = 1.3426 (sec), leaf count = 551
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [-\frac {6 \sqrt [6]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{2/3} \, _2F_1\left (\frac {1}{6},\frac {2}{3};\frac {7}{6};\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$1})^{2/3}}\& \right ]\left [\int _1^x-\sqrt [6]{f(K[1])}dK[1]+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [-\frac {6 \sqrt [6]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{2/3} \, _2F_1\left (\frac {1}{6},\frac {2}{3};\frac {7}{6};\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$1})^{2/3}}\& \right ]\left [\int _1^x\sqrt [6]{f(K[2])}dK[2]+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [-\frac {6 \sqrt [6]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{2/3} \, _2F_1\left (\frac {1}{6},\frac {2}{3};\frac {7}{6};\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$1})^{2/3}}\& \right ]\left [\int _1^x-\sqrt [3]{-1} \sqrt [6]{f(K[3])}dK[3]+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [-\frac {6 \sqrt [6]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{2/3} \, _2F_1\left (\frac {1}{6},\frac {2}{3};\frac {7}{6};\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$1})^{2/3}}\& \right ]\left [\int _1^x\sqrt [3]{-1} \sqrt [6]{f(K[4])}dK[4]+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [-\frac {6 \sqrt [6]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{2/3} \, _2F_1\left (\frac {1}{6},\frac {2}{3};\frac {7}{6};\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$1})^{2/3}}\& \right ]\left [\int _1^x-(-1)^{2/3} \sqrt [6]{f(K[5])}dK[5]+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [-\frac {6 \sqrt [6]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{2/3} \, _2F_1\left (\frac {1}{6},\frac {2}{3};\frac {7}{6};\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$1})^{2/3}}\& \right ]\left [\int _1^x(-1)^{2/3} \sqrt [6]{f(K[6])}dK[6]+c_1\right ]\right \}\right \}\]
Maple ✓
cpu = 0.874 (sec), leaf count = 68
\[\left [\int _{}^{y \left (x \right )}\frac {1}{\left (\textit {\_a} -a \right )^{\frac {5}{6}} \left (\textit {\_a} -b \right )^{\frac {2}{3}}}d \textit {\_a} +\int _{}^{x}-\frac {\left (f \left (\textit {\_a} \right ) \left (b -y \left (x \right )\right )^{4} \left (a -y \left (x \right )\right )^{5}\right )^{\frac {1}{6}}}{\left (y \left (x \right )-a \right )^{\frac {5}{6}} \left (y \left (x \right )-b \right )^{\frac {2}{3}}}d \textit {\_a} +\textit {\_C1} = 0\right ]\] Mathematica raw input
DSolve[f[x]*(-a + y[x])^5*(-b + y[x])^4 + y'[x]^6 == 0,y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[(-6*Hypergeometric2F1[1/6, 2/3, 7/6, (a - #1)/(a - b)]*(a - #1)^(1/6)*((-b + #1)/(a - b))^(2/3))/(b - #1)^(2/3) & ][C[1] + Inactive[Integrate][-f[K[1]]^(1/6), {K[1], 1, x}]]}, {y[x] -> InverseFunction[(-6*Hypergeometric2F1[1/6, 2/3, 7/6, (a - #1)/(a - b)]*(a - #1)^(1/6)*((-b + #1)/(a - b))^(2/3))/(b - #1)^(2/3) & ][C[1] + Inactive[Integrate][f[K[2]]^(1/6), {K[2], 1, x}]]}, {y[x] -> InverseFunction[(-6*Hypergeometric2F1[1/6, 2/3, 7/6, (a - #1)/(a - b)]*(a - #1)^(1/6)*((-b + #1)/(a - b))^(2/3))/(b - #1)^(2/3) & ][C[1] + Inactive[Integrate][-((-1)^(1/3)*f[K[3]]^(1/6)), {K[3], 1, x}]]}, {y[x] -> InverseFunction[(-6*Hypergeometric2F1[1/6, 2/3, 7/6, (a - #1)/(a - b)]*(a - #1)^(1/6)*((-b + #1)/(a - b))^(2/3))/(b - #1)^(2/3) & ][C[1] + Inactive[Integrate][(-1)^(1/3)*f[K[4]]^(1/6), {K[4], 1, x}]]}, {y[x] -> InverseFunction[(-6*Hypergeometric2F1[1/6, 2/3, 7/6, (a - #1)/(a - b)]*(a - #1)^(1/6)*((-b + #1)/(a - b))^(2/3))/(b - #1)^(2/3) & ][C[1] + Inactive[Integrate][-((-1)^(2/3)*f[K[5]]^(1/6)), {K[5], 1, x}]]}, {y[x] -> InverseFunction[(-6*Hypergeometric2F1[1/6, 2/3, 7/6, (a - #1)/(a - b)]*(a - #1)^(1/6)*((-b + #1)/(a - b))^(2/3))/(b - #1)^(2/3) & ][C[1] + Inactive[Integrate][(-1)^(2/3)*f[K[6]]^(1/6), {K[6], 1, x}]]}}
Maple raw input
dsolve(diff(y(x),x)^6+f(x)*(y(x)-a)^5*(y(x)-b)^4 = 0, y(x))
Maple raw output
[Intat(1/(_a-a)^(5/6)/(_a-b)^(2/3),_a = y(x))+Intat(-(f(_a)*(b-y(x))^4*(a-y(x))^5)^(1/6)/(y(x)-a)^(5/6)/(y(x)-b)^(2/3),_a = x)+_C1 = 0]