ODE
\[ f(x) (y(x)-a)^3 (y(x)-b)^2+y'(x)^4=0 \] 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 = 0.986594 (sec), leaf count = 415
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [-\frac {\sqrt [4]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} B_{\frac {a-\text {$\#$1}}{a-b}}\left (\frac {1}{4},\frac {1}{2}\right )}{\sqrt {b-\text {$\#$1}} \sqrt [4]{\frac {a-\text {$\#$1}}{a-b}}}\& \right ]\left [\int _1^x -\sqrt [4]{f(K[1])} \, dK[1]+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [-\frac {\sqrt [4]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} B_{\frac {a-\text {$\#$1}}{a-b}}\left (\frac {1}{4},\frac {1}{2}\right )}{\sqrt {b-\text {$\#$1}} \sqrt [4]{\frac {a-\text {$\#$1}}{a-b}}}\& \right ]\left [c_1+\int _1^x -i \sqrt [4]{f(K[2])} \, dK[2]\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [-\frac {\sqrt [4]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} B_{\frac {a-\text {$\#$1}}{a-b}}\left (\frac {1}{4},\frac {1}{2}\right )}{\sqrt {b-\text {$\#$1}} \sqrt [4]{\frac {a-\text {$\#$1}}{a-b}}}\& \right ]\left [c_1+\int _1^x i \sqrt [4]{f(K[3])} \, dK[3]\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [-\frac {\sqrt [4]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} B_{\frac {a-\text {$\#$1}}{a-b}}\left (\frac {1}{4},\frac {1}{2}\right )}{\sqrt {b-\text {$\#$1}} \sqrt [4]{\frac {a-\text {$\#$1}}{a-b}}}\& \right ]\left [\int _1^x \sqrt [4]{f(K[4])} \, dK[4]+c_1\right ]\right \}\right \}\]
Maple ✓
cpu = 0.345 (sec), leaf count = 270
\[ \left \{ \int ^{y \left ( x \right ) }\!{1{\frac {1}{\sqrt {{\it \_a}-b}}} \left ( {\it \_a}-a \right ) ^{-{\frac {3}{4}}}}{d{\it \_a}}+\int ^{x}\!{1\sqrt [4]{f \left ( {\it \_a} \right ) \left ( a-y \left ( x \right ) \right ) ^{3} \left ( b-y \left ( x \right ) \right ) ^{2}}{\frac {1}{\sqrt {y \left ( x \right ) -b}}} \left ( y \left ( x \right ) -a \right ) ^{-{\frac {3}{4}}}}{d{\it \_a}}+{\it \_C1}=0,\int ^{y \left ( x \right ) }\!{1{\frac {1}{\sqrt {{\it \_a}-b}}} \left ( {\it \_a}-a \right ) ^{-{\frac {3}{4}}}}{d{\it \_a}}+\int ^{x}\!{-i\sqrt [4]{f \left ( {\it \_a} \right ) \left ( a-y \left ( x \right ) \right ) ^{3} \left ( b-y \left ( x \right ) \right ) ^{2}}{\frac {1}{\sqrt {y \left ( x \right ) -b}}} \left ( y \left ( x \right ) -a \right ) ^{-{\frac {3}{4}}}}{d{\it \_a}}+{\it \_C1}=0,\int ^{y \left ( x \right ) }\!{1{\frac {1}{\sqrt {{\it \_a}-b}}} \left ( {\it \_a}-a \right ) ^{-{\frac {3}{4}}}}{d{\it \_a}}+\int ^{x}\!{i\sqrt [4]{f \left ( {\it \_a} \right ) \left ( a-y \left ( x \right ) \right ) ^{3} \left ( b-y \left ( x \right ) \right ) ^{2}}{\frac {1}{\sqrt {y \left ( x \right ) -b}}} \left ( y \left ( x \right ) -a \right ) ^{-{\frac {3}{4}}}}{d{\it \_a}}+{\it \_C1}=0,\int ^{y \left ( x \right ) }\!{1{\frac {1}{\sqrt {{\it \_a}-b}}} \left ( {\it \_a}-a \right ) ^{-{\frac {3}{4}}}}{d{\it \_a}}+\int ^{x}\!-{1\sqrt [4]{f \left ( {\it \_a} \right ) \left ( a-y \left ( x \right ) \right ) ^{3} \left ( b-y \left ( x \right ) \right ) ^{2}}{\frac {1}{\sqrt {y \left ( x \right ) -b}}} \left ( y \left ( x \right ) -a \right ) ^{-{\frac {3}{4}}}}{d{\it \_a}}+{\it \_C1}=0 \right \} \] Mathematica raw input
DSolve[f[x]*(-a + y[x])^3*(-b + y[x])^2 + y'[x]^4 == 0,y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[-((Beta[(a - #1)/(a - b), 1/4, 1/2]*(a - #1)^(1/4)*Sqrt[(-b + #1)/(a - b)])/(((a - #1)/(a - b))^(1/4)*Sqrt[b - #1])) & ][C[1] + Integrate[-f[K[1]]^(1/4), {K[1], 1, x}]]}, {y[x] -> InverseFunction[-((Beta[(a - #1)/(a - b), 1/4, 1/2]*(a - #1)^(1/4)*Sqrt[(-b + #1)/(a - b)])/(((a - #1)/(a - b))^(1/4)*Sqrt[b - #1])) & ][C[1] + Integrate[(-I)*f[K[2]]^(1/4), {K[2], 1, x}]]}, {y[x] -> InverseFunction[-((Beta[(a - #1)/(a - b), 1/4, 1/2]*(a - #1)^(1/4)*Sqrt[(-b + #1)/(a - b)])/(((a - #1)/(a - b))^(1/4)*Sqrt[b - #1])) & ][C[1] + Integrate[I*f[K[3]]^(1/4), {K[3], 1, x}]]}, {y[x] -> InverseFunction[-((Beta[(a - #1)/(a - b), 1/4, 1/2]*(a - #1)^(1/4)*Sqrt[(-b + #1)/(a - b)])/(((a - #1)/(a - b))^(1/4)*Sqrt[b - #1])) & ][C[1] + Integrate[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)^2 = 0, y(x),'implicit')
Maple raw output
Intat(1/(_a-b)^(1/2)/(_a-a)^(3/4),_a = y(x))+Intat(-(f(_a)*(a-y(x))^3*(b-y(x))^2)^(1/4)/(y(x)-b)^(1/2)/(y(x)-a)^(3/4),_a = x)+_C1 = 0, Intat(1/(_a-b)^(1/2)/(_a-a)^(3/4),_a = y(x))+Intat(I*(f(_a)*(a-y(x))^3*(b-y(x))^2)^(1/4)/(y(x)-b)^(1/2)/(y(x)-a)^(3/4),_a = x)+_C1 = 0, Intat(1/(_a-b)^(1/2)/(_a-a)^(3/4),_a = y(x))+Intat(-I*(f(_a)*(a-y(x))^3*(b-y(x))^2)^(1/4)/(y(x)-b)^(1/2)/(y(x)-a)^(3/4),_a = x)+_C1 = 0, Intat(1/(_a-b)^(1/2)/(_a-a)^(3/4),_a = y(x))+Intat((f(_a)*(a-y(x))^3*(b-y(x))^2)^(1/4)/(y(x)-b)^(1/2)/(y(x)-a)^(3/4),_a = x)+_C1 = 0