29.25.24 problem 721

Internal problem ID [5311]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 25
Problem number : 721
Date solved : Monday, January 27, 2025 at 11:08:25 AM
CAS classification : [_separable]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 67

dsolve(diff(y(x),x)*sqrt(b^2+y(x)^2) = sqrt(a^2+x^2),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.906 (sec). Leaf size: 89

DSolve[D[y[x],x] Sqrt[y[x]^2+b^2]==Sqrt[x^2+a^2],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \text {InverseFunction}\left [\frac {1}{2} \text {$\#$1} \sqrt {\text {$\#$1}^2+b^2}+\frac {1}{2} b^2 \log \left (\sqrt {\text {$\#$1}^2+b^2}+\text {$\#$1}\right )\&\right ]\left [\frac {1}{2} x \sqrt {a^2+x^2}+\frac {1}{2} a^2 \log \left (\sqrt {a^2+x^2}+x\right )+c_1\right ] \]