77.22.16 problem 16

Internal problem ID [20555]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter IV. Equations of the first order but not of the first degree. Exercise IV (E) at page 63
Problem number : 16
Date solved : Thursday, October 02, 2025 at 06:06:28 PM
CAS classification : [_dAlembert]

\begin{align*} y y^{\prime }+x&=a {y^{\prime }}^{2} \end{align*}

✓ Maple. Time used: 0.044 (sec). Leaf size: 246

ode:=x+y(x)*diff(y(x),x) = a*diff(y(x),x)^2; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} \frac {-\frac {\sqrt {2}\, \left (y+\sqrt {4 a x +y^{2}}\right ) \operatorname {arcsinh}\left (\frac {y+\sqrt {4 a x +y^{2}}}{2 a}\right )}{2}+x \sqrt {\frac {y \sqrt {4 a x +y^{2}}+2 a^{2}+2 a x +y^{2}}{a^{2}}}+c_1 y+c_1 \sqrt {4 a x +y^{2}}}{\sqrt {\frac {y \sqrt {4 a x +y^{2}}+y^{2}+2 a \left (a +x \right )}{a^{2}}}} &= 0 \\ \frac {x \sqrt {\frac {-2 y \sqrt {4 a x +y^{2}}+2 y^{2}+4 a \left (a +x \right )}{a^{2}}}-\left (y-\sqrt {4 a x +y^{2}}\right ) \left (-\operatorname {arcsinh}\left (\frac {-y+\sqrt {4 a x +y^{2}}}{2 a}\right )+c_1 \right )}{\sqrt {\frac {-2 y \sqrt {4 a x +y^{2}}+2 y^{2}+4 a \left (a +x \right )}{a^{2}}}} &= 0 \\ \end{align*}

✓ Mathematica. Time used: 0.316 (sec). Leaf size: 57

ode=x+y[x]*D[y[x],x]==a*D[y[x],x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\left \{x=\frac {a K[1] \text {arcsinh}(K[1])}{\sqrt {K[1]^2+1}}+\frac {c_1 K[1]}{\sqrt {K[1]^2+1}},y(x)=a K[1]-\frac {x}{K[1]}\right \},\{y(x),K[1]\}\right ] \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-a*Derivative(y(x), x)**2 + x + y(x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out