60.1.373 problem 382
Internal
problem
ID
[10387]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
382
Date
solved
:
Wednesday, March 05, 2025 at 10:43:12 AM
CAS
classification
:
[_quadrature]
\begin{align*} {y^{\prime }}^{2}+a x y^{\prime }-b \,x^{2}-c&=0 \end{align*}
✓ Maple. Time used: 0.028 (sec). Leaf size: 174
ode:=diff(y(x),x)^2+a*x*diff(y(x),x)-b*x^2-c = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -\frac {4 c \ln \left (\sqrt {a^{2}+4 b}\, x +\sqrt {\left (a^{2}+4 b \right ) x^{2}+4 c}\right )+\sqrt {a^{2}+4 b}\, \left (a \,x^{2}+x \sqrt {\left (a^{2}+4 b \right ) x^{2}+4 c}-4 c_{1} \right )}{4 \sqrt {a^{2}+4 b}} \\
y &= -\frac {-4 c \ln \left (\sqrt {a^{2}+4 b}\, x +\sqrt {\left (a^{2}+4 b \right ) x^{2}+4 c}\right )+\sqrt {a^{2}+4 b}\, \left (a \,x^{2}-x \sqrt {\left (a^{2}+4 b \right ) x^{2}+4 c}-4 c_{1} \right )}{4 \sqrt {a^{2}+4 b}} \\
\end{align*}
✓ Mathematica. Time used: 0.49 (sec). Leaf size: 213
ode=-c - b*x^2 + a*x*D[y[x],x] + D[y[x],x]^2==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {c^{3/2} \sqrt {\frac {x^2 \left (a^2+4 b\right )}{c}+4} \arcsin \left (\frac {x \sqrt {-a^2-4 b}}{2 \sqrt {c}}\right )}{\sqrt {-a^2-4 b} \sqrt {x^2 \left (a^2+4 b\right )+4 c}}-\frac {1}{4} x \left (\sqrt {x^2 \left (a^2+4 b\right )+4 c}+a x\right )+c_1 \\
y(x)\to \frac {1}{4} x \sqrt {x^2 \left (a^2+4 b\right )+4 c}+\frac {c \log \left (\sqrt {a^2+4 b} \sqrt {x^2 \left (a^2+4 b\right )+4 c}+a^2 x+4 b x\right )}{\sqrt {a^2+4 b}}-\frac {a x^2}{4}+c_1 \\
\end{align*}
✓ Sympy. Time used: 0.872 (sec). Leaf size: 303
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
c = symbols("c")
y = Function("y")
ode = Eq(a*x*Derivative(y(x), x) - b*x**2 - c + Derivative(y(x), x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = C_{1} - \frac {a x^{2}}{4} - \frac {\begin {cases} \frac {x \left (a^{2} + 4 b\right ) \sqrt {4 c + x^{2} \left (a^{2} + 4 b\right )}}{2 a^{2} + 8 b} + \left (- \frac {4 c \left (a^{2} + 4 b\right )}{2 a^{2} + 8 b} + 4 c\right ) \left (\begin {cases} \frac {\log {\left (x \left (2 a^{2} + 8 b\right ) + 2 \sqrt {a^{2} + 4 b} \sqrt {4 c + x^{2} \left (a^{2} + 4 b\right )} \right )}}{\sqrt {a^{2} + 4 b}} & \text {for}\: c \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {x^{2} \left (a^{2} + 4 b\right )}} & \text {otherwise} \end {cases}\right ) & \text {for}\: a^{2} + 4 b \neq 0 \\2 \sqrt {c} x & \text {otherwise} \end {cases}}{2}, \ y{\left (x \right )} = C_{1} - \frac {a x^{2}}{4} + \frac {\begin {cases} \frac {x \left (a^{2} + 4 b\right ) \sqrt {4 c + x^{2} \left (a^{2} + 4 b\right )}}{2 a^{2} + 8 b} + \left (- \frac {4 c \left (a^{2} + 4 b\right )}{2 a^{2} + 8 b} + 4 c\right ) \left (\begin {cases} \frac {\log {\left (x \left (2 a^{2} + 8 b\right ) + 2 \sqrt {a^{2} + 4 b} \sqrt {4 c + x^{2} \left (a^{2} + 4 b\right )} \right )}}{\sqrt {a^{2} + 4 b}} & \text {for}\: c \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {x^{2} \left (a^{2} + 4 b\right )}} & \text {otherwise} \end {cases}\right ) & \text {for}\: a^{2} + 4 b \neq 0 \\2 \sqrt {c} x & \text {otherwise} \end {cases}}{2}\right ]
\]