62.12.25 problem Ex 26

Internal problem ID [12871]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter 2, differential equations of the first order and the first degree. Article 19. Summary. Page 29
Problem number : Ex 26
Date solved : Tuesday, January 28, 2025 at 04:29:45 AM
CAS classification : [[_1st_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}+y^{2}\right ) \left (x +y^{\prime } y\right )+\sqrt {1+x^{2}+y^{2}}\, \left (y-x y^{\prime }\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.266 (sec). Leaf size: 25

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

\[ \arctan \left (\frac {x}{y}\right )+\sqrt {x^{2}+y^{2}+1}-c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 0.421 (sec). Leaf size: 661

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

\[ \text {Solve}\left [\int _1^{y(x)}\left (\exp \left (\int _1^{x^2+K[3]^2}\left (-\frac {1}{2 (K[1]+1)}-\frac {1}{K[1]}\right )dK[1]\right ) K[3]^3+\exp \left (\int _1^{x^2+K[3]^2}\left (-\frac {1}{2 (K[1]+1)}-\frac {1}{K[1]}\right )dK[1]\right ) x^2 K[3]-\int _1^x\left (2 \exp \left (\int _1^{K[2]^2+K[3]^2}\left (-\frac {1}{2 (K[1]+1)}-\frac {1}{K[1]}\right )dK[1]\right ) K[3] \left (-\frac {1}{2 \left (K[2]^2+K[3]^2+1\right )}-\frac {1}{K[2]^2+K[3]^2}\right ) K[2]^3+2 \exp \left (\int _1^{K[2]^2+K[3]^2}\left (-\frac {1}{2 (K[1]+1)}-\frac {1}{K[1]}\right )dK[1]\right ) K[3] K[2]+2 \exp \left (\int _1^{K[2]^2+K[3]^2}\left (-\frac {1}{2 (K[1]+1)}-\frac {1}{K[1]}\right )dK[1]\right ) K[3]^3 \left (-\frac {1}{2 \left (K[2]^2+K[3]^2+1\right )}-\frac {1}{K[2]^2+K[3]^2}\right ) K[2]+2 \exp \left (\int _1^{K[2]^2+K[3]^2}\left (-\frac {1}{2 (K[1]+1)}-\frac {1}{K[1]}\right )dK[1]\right ) K[3]^2 \sqrt {K[2]^2+K[3]^2+1} \left (-\frac {1}{2 \left (K[2]^2+K[3]^2+1\right )}-\frac {1}{K[2]^2+K[3]^2}\right )+\exp \left (\int _1^{K[2]^2+K[3]^2}\left (-\frac {1}{2 (K[1]+1)}-\frac {1}{K[1]}\right )dK[1]\right ) \sqrt {K[2]^2+K[3]^2+1}+\frac {\exp \left (\int _1^{K[2]^2+K[3]^2}\left (-\frac {1}{2 (K[1]+1)}-\frac {1}{K[1]}\right )dK[1]\right ) K[3]^2}{\sqrt {K[2]^2+K[3]^2+1}}\right )dK[2]-\exp \left (\int _1^{x^2+K[3]^2}\left (-\frac {1}{2 (K[1]+1)}-\frac {1}{K[1]}\right )dK[1]\right ) x \sqrt {x^2+K[3]^2+1}\right )dK[3]+\int _1^x\left (\exp \left (\int _1^{K[2]^2+y(x)^2}\left (-\frac {1}{2 (K[1]+1)}-\frac {1}{K[1]}\right )dK[1]\right ) K[2]^3+\exp \left (\int _1^{K[2]^2+y(x)^2}\left (-\frac {1}{2 (K[1]+1)}-\frac {1}{K[1]}\right )dK[1]\right ) y(x)^2 K[2]+\exp \left (\int _1^{K[2]^2+y(x)^2}\left (-\frac {1}{2 (K[1]+1)}-\frac {1}{K[1]}\right )dK[1]\right ) y(x) \sqrt {K[2]^2+y(x)^2+1}\right )dK[2]=c_1,y(x)\right ] \]