82.18.7 problem Ex. 8

Internal problem ID [18829]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter III. Equations of the first order but not of the first degree. Examples on chapter III. page 38
Problem number : Ex. 8
Date solved : Tuesday, January 28, 2025 at 12:23:04 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

\begin{align*} \left (-y+x y^{\prime }\right )^{2}&=a \left ({y^{\prime }}^{2}+1\right ) \left (x^{2}+y^{2}\right )^{{3}/{2}} \end{align*}

✗ Solution by Maple

dsolve((x*diff(y(x),x)-y(x))^2=a*(1+diff(y(x),x)^2)*(x^2+y(x)^2)^(3/2),y(x), singsol=all)
 

\[ \text {No solution found} \]

✓ Solution by Mathematica

Time used: 67.315 (sec). Leaf size: 305

DSolve[(x*D[y[x],x]-y[x])^2==a*(1+D[y[x],x]^2)*(x^2+y[x]^2)^(3/2),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} \text {Solve}\left [\arctan \left (\frac {x}{y(x)}\right )-\frac {2 \sqrt {a \left (x^2+y(x)^2\right )^2 \left (-a x^2-a y(x)^2+\sqrt {x^2+y(x)^2}\right )} \arctan \left (\frac {\sqrt {-a x^2-a y(x)^2+\sqrt {x^2+y(x)^2}}}{\sqrt {a} \sqrt {x^2+y(x)^2}}\right )}{\sqrt {a} \left (x^2+y(x)^2\right ) \sqrt {-a x^2-a y(x)^2+\sqrt {x^2+y(x)^2}}}&=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {2 \sqrt {a \left (x^2+y(x)^2\right )^2 \left (-a x^2-a y(x)^2+\sqrt {x^2+y(x)^2}\right )} \arctan \left (\frac {\sqrt {-a x^2-a y(x)^2+\sqrt {x^2+y(x)^2}}}{\sqrt {a} \sqrt {x^2+y(x)^2}}\right )}{\sqrt {a} \left (x^2+y(x)^2\right ) \sqrt {-a x^2-a y(x)^2+\sqrt {x^2+y(x)^2}}}+\arctan \left (\frac {x}{y(x)}\right )&=c_1,y(x)\right ] \\ \end{align*}