60.1.475 problem 478

Internal problem ID [10489]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 478
Date solved : Monday, January 27, 2025 at 07:55:20 PM
CAS classification : [_quadrature]

\begin{align*} \left (a y+b \right ) \left (1+{y^{\prime }}^{2}\right )-c&=0 \end{align*}

✓ Solution by Maple

Time used: 0.049 (sec). Leaf size: 162

dsolve((a*y(x)+b)*(diff(y(x),x)^2+1)-c = 0,y(x), singsol=all)
 

\begin{align*} y &= \frac {c -b}{a} \\ \frac {-\arctan \left (\frac {2 a y+2 b -c}{2 \sqrt {-\left (a y+b \right ) \left (a y+b -c \right )}}\right ) c +2 \sqrt {-\left (a y+b \right ) \left (a y+b -c \right )}+\left (-2 c_{1} +2 x \right ) a}{2 a} &= 0 \\ \frac {\arctan \left (\frac {2 a y+2 b -c}{2 \sqrt {-\left (a y+b \right ) \left (a y+b -c \right )}}\right ) c -2 \sqrt {-\left (a y+b \right ) \left (a y+b -c \right )}+\left (-2 c_{1} +2 x \right ) a}{2 a} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.679 (sec). Leaf size: 240

DSolve[-c + (b + a*y[x])*(1 + D[y[x],x]^2)==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \text {InverseFunction}\left [\frac {-\sqrt {-a} c \sqrt {-a c} \sqrt {\frac {\text {$\#$1} a+b}{c}} \arcsin \left (\frac {a \sqrt {-\text {$\#$1} a-b+c}}{\sqrt {-a} \sqrt {-a c}}\right )-a (\text {$\#$1} a+b) \sqrt {-\text {$\#$1} a-b+c}}{a^2 \sqrt {\text {$\#$1} a+b}}\&\right ][-x+c_1] \\ y(x)\to \text {InverseFunction}\left [\frac {-\sqrt {-a} c \sqrt {-a c} \sqrt {\frac {\text {$\#$1} a+b}{c}} \arcsin \left (\frac {a \sqrt {-\text {$\#$1} a-b+c}}{\sqrt {-a} \sqrt {-a c}}\right )-a (\text {$\#$1} a+b) \sqrt {-\text {$\#$1} a-b+c}}{a^2 \sqrt {\text {$\#$1} a+b}}\&\right ][x+c_1] \\ y(x)\to \frac {c-b}{a} \\ \end{align*}