22.1.81 problem 84

Internal problem ID [4387]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 2. First-Order and Simple Higher-Order Differential Equations. Page 78
Problem number : 84
Date solved : Tuesday, September 30, 2025 at 07:24:03 AM
CAS classification : [_quadrature]

\begin{align*} x&=y^{\prime } \sqrt {{y^{\prime }}^{2}+1} \end{align*}

✓ Maple. Time used: 0.064 (sec). Leaf size: 143

ode:=x = diff(y(x),x)*(diff(y(x),x)^2+1)^(1/2); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -\frac {i \left (-32 x^{4}-4 x^{2}+1\right ) \sinh \left (\frac {3 \,\operatorname {arcsinh}\left (2 x \right )}{2}\right )}{3 \sqrt {4 x^{2}+1}}-\frac {16 i x^{3} \cosh \left (\frac {3 \,\operatorname {arcsinh}\left (2 x \right )}{2}\right )}{3}+c_1 \\ y &= \frac {i \left (-32 x^{4}-4 x^{2}+1\right ) \sinh \left (\frac {3 \,\operatorname {arcsinh}\left (2 x \right )}{2}\right )}{3 \sqrt {4 x^{2}+1}}+\frac {16 i x^{3} \cosh \left (\frac {3 \,\operatorname {arcsinh}\left (2 x \right )}{2}\right )}{3}+c_1 \\ y &= -\frac {\int \sqrt {-2+2 \sqrt {4 x^{2}+1}}d x}{2}+c_1 \\ y &= \frac {\int \sqrt {-2+2 \sqrt {4 x^{2}+1}}d x}{2}+c_1 \\ \end{align*}

✓ Mathematica. Time used: 1.56 (sec). Leaf size: 207

ode=x==D[y[x],x]*Sqrt[ (D[y[x],x])^2+1]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {\sqrt {2} x \left (\sqrt {4 x^2+1}-2\right )}{3 \sqrt {\sqrt {4 x^2+1}-1}}+c_1\\ y(x)&\to \frac {\sqrt {2} x \left (\sqrt {4 x^2+1}-2\right )}{3 \sqrt {\sqrt {4 x^2+1}-1}}+c_1\\ y(x)&\to -\frac {\sqrt {2} x \left (4 x^2+3 \sqrt {4 x^2+1}+3\right )}{3 \left (-\sqrt {4 x^2+1}-1\right )^{3/2}}+c_1\\ y(x)&\to \frac {\sqrt {2} x \left (4 x^2+3 \sqrt {4 x^2+1}+3\right )}{3 \left (-\sqrt {4 x^2+1}-1\right )^{3/2}}+c_1 \end{align*}

✓ Sympy. Time used: 2.577 (sec). Leaf size: 966

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x - sqrt(Derivative(y(x), x)**2 + 1)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \text {Solution too large to show} \]