60.1.498 problem 501
Internal
problem
ID
[10512]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
501
Date
solved
:
Monday, January 27, 2025 at 08:31:58 PM
CAS
classification
:
[`y=_G(x,y')`]
\begin{align*} \left (a y^{2}+b x +c \right ) {y^{\prime }}^{2}-b y y^{\prime }+d y^{2}&=0 \end{align*}
✓ Solution by Maple
Time used: 4.089 (sec). Leaf size: 50
dsolve((a*y(x)^2+b*x+c)*diff(y(x),x)^2-b*y(x)*diff(y(x),x)+d*y(x)^2=0,y(x), singsol=all)
\begin{align*}
y &= 0 \\
y &= \frac {\left (-b x -c \right ) \sqrt {-a d}}{a b} \\
y &= \frac {\sqrt {-a d}\, \left (b x +c \right )}{a b} \\
\end{align*}
✓ Solution by Mathematica
Time used: 69.621 (sec). Leaf size: 980
DSolve[d*y[x]^2 - b*y[x]*D[y[x],x] + (c + b*x + a*y[x]^2)*D[y[x],x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
\text {Solve}\left [\left \{y(x)&=\frac {b K[1]-\sqrt {-K[1]^2 \left (4 a b x K[1]^2+4 a c K[1]^2-b^2+4 b d x+4 c d\right )}}{2 \left (a K[1]^2+d\right )},x=\frac {-2 b^2 c_1 d^{5/2} \log \left (\sqrt {d} \sqrt {a K[1]^2+d}+d\right )+2 a b^2 c_1 d^{3/2} K[1]^2 \log (K[1])-2 a b^2 c_1 d^{3/2} K[1]^2 \log \left (\sqrt {d} \sqrt {a K[1]^2+d}+d\right )-a b^2 c_1{}^2 d^3 K[1]^2+2 b^2 c_1 d^2 \sqrt {a K[1]^2+d}-b^2 d \log ^2\left (\sqrt {d} \sqrt {a K[1]^2+d}+d\right )-a b^2 K[1]^2 \log ^2\left (\sqrt {d} \sqrt {a K[1]^2+d}+d\right )+2 b^2 d \log (K[1]) \log \left (\sqrt {d} \sqrt {a K[1]^2+d}+d\right )-2 b^2 \sqrt {d} \log (K[1]) \sqrt {a K[1]^2+d}+2 b^2 \sqrt {d} \sqrt {a K[1]^2+d} \log \left (\sqrt {d} \sqrt {a K[1]^2+d}+d\right )+2 a b^2 K[1]^2 \log (K[1]) \log \left (\sqrt {d} \sqrt {a K[1]^2+d}+d\right )-a b^2 K[1]^2 \log ^2(K[1])-4 a c d K[1]^2+2 b^2 c_1 d^{5/2} \log (K[1])-b^2 d \log ^2(K[1])-b^2 c_1{}^2 d^4-4 c d^2}{4 b d \left (a K[1]^2+d\right )}\right \},\{y(x),K[1]\}\right ] \\
\text {Solve}\left [\left \{y(x)&=\frac {\sqrt {-K[2]^2 \left (4 a b x K[2]^2+4 a c K[2]^2-b^2+4 b d x+4 c d\right )}+b K[2]}{2 \left (a K[2]^2+d\right )},x=\frac {-2 b^2 c_1 d^{5/2} \log \left (\sqrt {d} \sqrt {a K[2]^2+d}+d\right )+2 a b^2 c_1 d^{3/2} K[2]^2 \log (K[2])-2 a b^2 c_1 d^{3/2} K[2]^2 \log \left (\sqrt {d} \sqrt {a K[2]^2+d}+d\right )-a b^2 c_1{}^2 d^3 K[2]^2+2 b^2 c_1 d^2 \sqrt {a K[2]^2+d}-b^2 d \log ^2\left (\sqrt {d} \sqrt {a K[2]^2+d}+d\right )-a b^2 K[2]^2 \log ^2\left (\sqrt {d} \sqrt {a K[2]^2+d}+d\right )+2 b^2 d \log (K[2]) \log \left (\sqrt {d} \sqrt {a K[2]^2+d}+d\right )-2 b^2 \sqrt {d} \log (K[2]) \sqrt {a K[2]^2+d}+2 b^2 \sqrt {d} \sqrt {a K[2]^2+d} \log \left (\sqrt {d} \sqrt {a K[2]^2+d}+d\right )+2 a b^2 K[2]^2 \log (K[2]) \log \left (\sqrt {d} \sqrt {a K[2]^2+d}+d\right )-a b^2 K[2]^2 \log ^2(K[2])-4 a c d K[2]^2+2 b^2 c_1 d^{5/2} \log (K[2])-b^2 d \log ^2(K[2])-b^2 c_1{}^2 d^4-4 c d^2}{4 b d \left (a K[2]^2+d\right )}\right \},\{y(x),K[2]\}\right ] \\
y(x)\to 0 \\
\end{align*}