29.33.24 problem 987

Internal problem ID [5563]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 33
Problem number : 987
Date solved : Monday, January 27, 2025 at 12:00:39 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 0.184 (sec). Leaf size: 77

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

\begin{align*} y \left (x \right ) &= -i x \\ y \left (x \right ) &= i x \\ y \left (x \right ) &= \tan \left (\operatorname {RootOf}\left (-2 \textit {\_Z} \sqrt {a^{2}-1}-\ln \left (\sec \left (\textit {\_Z} \right )^{2} x^{2}\right )+2 c_{1} \right )\right ) x \\ y \left (x \right ) &= \tan \left (\operatorname {RootOf}\left (2 \textit {\_Z} \sqrt {a^{2}-1}-\ln \left (\sec \left (\textit {\_Z} \right )^{2} x^{2}\right )+2 c_{1} \right )\right ) x \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.405 (sec). Leaf size: 115

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

\begin{align*} \text {Solve}\left [\sqrt {a-1} \sqrt {a+1} \arctan \left (\frac {y(x)}{x}\right )-\frac {1}{2} \log \left (\frac {y(x)^2}{x^2}+1\right )&=\log (x)+c_1,y(x)\right ] \\ \text {Solve}\left [\sqrt {a-1} \sqrt {a+1} \arctan \left (\frac {y(x)}{x}\right )+\frac {1}{2} \log \left (\frac {y(x)^2}{x^2}+1\right )&=-\log (x)+c_1,y(x)\right ] \\ y(x)\to -i x \\ y(x)\to i x \\ \end{align*}