ODE
\[ 4 y(x) y''(x)=a y(x)+b y(x)^2+c y(x)^3+3 y'(x)^2 \] ODE Classification
[[_2nd_order, _missing_x]]
Book solution method
TO DO
Mathematica ✓
cpu = 4.46683 (sec), leaf count = 2281
\[\left \{\text {Solve}\left [-\frac {4 F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,2\right ]-\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,4\right ]\right ) \left (\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,1\right ]-\sqrt {y(x)}\right )}{\left (\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,1\right ]-\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,4\right ]\right ) \left (\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,2\right ]-\sqrt {y(x)}\right )}}\right )|\frac {\left (\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,2\right ]-\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,3\right ]\right ) \left (\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,1\right ]-\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,4\right ]\right )}{\left (\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,1\right ]-\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,3\right ]\right ) \left (\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,2\right ]-\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,4\right ]\right )}\right ) \left (\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,1\right ]-\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,4\right ]\right ) \left (\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,2\right ]-\sqrt {y(x)}\right ){}^2 \sqrt {\frac {\left (\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,1\right ]-\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,2\right ]\right ) \left (\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,3\right ]-\sqrt {y(x)}\right )}{\left (\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,1\right ]-\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,3\right ]\right ) \left (\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,2\right ]-\sqrt {y(x)}\right )}} \sqrt {\frac {\left (\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,1\right ]-\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,2\right ]\right ) \left (\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,2\right ]-\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,4\right ]\right ) \left (\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,1\right ]-\sqrt {y(x)}\right ) \left (\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,4\right ]-\sqrt {y(x)}\right )}{\left (\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,1\right ]-\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,4\right ]\right ){}^2 \left (\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,2\right ]-\sqrt {y(x)}\right ){}^2}} \sqrt {y(x)}}{\left (\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,2\right ]-\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,1\right ]\right ) \left (\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,2\right ]-\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,4\right ]\right ) \sqrt {\frac {1}{3} c y(x)^3+b y(x)^2+c_1 y(x)^{3/2}-a y(x)}}=x+c_2,y(x)\right ],\text {Solve}\left [\frac {4 F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,2\right ]-\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,4\right ]\right ) \left (\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,1\right ]-\sqrt {y(x)}\right )}{\left (\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,1\right ]-\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,4\right ]\right ) \left (\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,2\right ]-\sqrt {y(x)}\right )}}\right )|\frac {\left (\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,2\right ]-\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,3\right ]\right ) \left (\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,1\right ]-\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,4\right ]\right )}{\left (\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,1\right ]-\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,3\right ]\right ) \left (\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,2\right ]-\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,4\right ]\right )}\right ) \left (\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,1\right ]-\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,4\right ]\right ) \left (\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,2\right ]-\sqrt {y(x)}\right ){}^2 \sqrt {\frac {\left (\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,1\right ]-\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,2\right ]\right ) \left (\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,3\right ]-\sqrt {y(x)}\right )}{\left (\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,1\right ]-\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,3\right ]\right ) \left (\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,2\right ]-\sqrt {y(x)}\right )}} \sqrt {\frac {\left (\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,1\right ]-\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,2\right ]\right ) \left (\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,2\right ]-\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,4\right ]\right ) \left (\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,1\right ]-\sqrt {y(x)}\right ) \left (\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,4\right ]-\sqrt {y(x)}\right )}{\left (\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,1\right ]-\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,4\right ]\right ){}^2 \left (\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,2\right ]-\sqrt {y(x)}\right ){}^2}} \sqrt {y(x)}}{\left (\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,2\right ]-\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,1\right ]\right ) \left (\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,2\right ]-\text {Root}\left [c \text {$\#$1}^4+3 b \text {$\#$1}^2+3 c_1 \text {$\#$1}-3 a\& ,4\right ]\right ) \sqrt {\frac {1}{3} c y(x)^3+b y(x)^2+c_1 y(x)^{3/2}-a y(x)}}=x+c_2,y(x)\right ]\right \}\]
Maple ✓
cpu = 0.53 (sec), leaf count = 87
\[ \left \{ \int ^{y \left ( x \right ) }\!-3\,{\frac {1}{\sqrt {9\,{\it \_C1}\,{{\it \_a}}^{3/2}+3\,c{{\it \_a}}^{3}+9\,{{\it \_a}}^{2}b-9\,{\it \_a}\,a}}}{d{\it \_a}}-x-{\it \_C2}=0,\int ^{y \left ( x \right ) }\!3\,{\frac {1}{\sqrt {9\,{\it \_C1}\,{{\it \_a}}^{3/2}+3\,c{{\it \_a}}^{3}+9\,{{\it \_a}}^{2}b-9\,{\it \_a}\,a}}}{d{\it \_a}}-x-{\it \_C2}=0,y \left ( x \right ) =0 \right \} \] Mathematica raw input
DSolve[4*y[x]*y''[x] == a*y[x] + b*y[x]^2 + c*y[x]^3 + 3*y'[x]^2,y[x],x]
Mathematica raw output
{Solve[(-4*EllipticF[ArcSin[Sqrt[((Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 2] - Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 4])*(Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 1] - Sqrt[y[x]]))/((Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 1] - Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 4])*(Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 2] - Sqrt[y[x]]))]], ((Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 2] - Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 3])*(Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 1] - Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 4]))/((Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 1] - Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 3])*(Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 2] - Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 4]))]*(Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 1] - Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 4])*(Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 2] - Sqrt[y[x]])^2*Sqrt[((Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 1] - Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 2])*(Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 3] - Sqrt[y[x]]))/((Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 1] - Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 3])*(Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 2] - Sqrt[y[x]]))]*Sqrt[((Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 1] - Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 2])*(Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 2] - Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 4])*(Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 1] - Sqrt[y[x]])*(Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 4] - Sqrt[y[x]]))/((Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 1] - Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 4])^2*(Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 2] - Sqrt[y[x]])^2)]*Sqrt[y[x]])/((-Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 1] + Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 2])*(Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 2] - Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 4])*Sqrt[-(a*y[x]) + C[1]*y[x]^(3/2) + b*y[x]^2 + (c*y[x]^3)/3]) == x + C[2], y[x]], Solve[(4*EllipticF[ArcSin[Sqrt[((Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 2] - Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 4])*(Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 1] - Sqrt[y[x]]))/((Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 1] - Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 4])*(Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 2] - Sqrt[y[x]]))]], ((Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 2] - Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 3])*(Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 1] - Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 4]))/((Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 1] - Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 3])*(Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 2] - Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 4]))]*(Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 1] - Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 4])*(Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 2] - Sqrt[y[x]])^2*Sqrt[((Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 1] - Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 2])*(Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 3] - Sqrt[y[x]]))/((Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 1] - Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 3])*(Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 2] - Sqrt[y[x]]))]*Sqrt[((Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 1] - Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 2])*(Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 2] - Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 4])*(Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 1] - Sqrt[y[x]])*(Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 4] - Sqrt[y[x]]))/((Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 1] - Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 4])^2*(Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 2] - Sqrt[y[x]])^2)]*Sqrt[y[x]])/((-Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 1] + Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 2])*(Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 2] - Root[-3*a + 3*C[1]*#1 + 3*b*#1^2 + c*#1^4 & , 4])*Sqrt[-(a*y[x]) + C[1]*y[x]^(3/2) + b*y[x]^2 + (c*y[x]^3)/3]) == x + C[2], y[x]]}
Maple raw input
dsolve(4*y(x)*diff(diff(y(x),x),x) = 3*diff(y(x),x)^2+a*y(x)+b*y(x)^2+c*y(x)^3, y(x),'implicit')
Maple raw output
y(x) = 0, Intat(-3/(9*_C1*_a^(3/2)+3*c*_a^3+9*_a^2*b-9*_a*a)^(1/2),_a = y(x))-x-_C2 = 0, Intat(3/(9*_C1*_a^(3/2)+3*c*_a^3+9*_a^2*b-9*_a*a)^(1/2),_a = y(x))-x-_C2 = 0