60.1.492 problem 495

Internal problem ID [10506]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 495
Date solved : Monday, January 27, 2025 at 08:25:56 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

✓ Solution by Maple

Time used: 0.164 (sec). Leaf size: 73

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

\begin{align*} y &= -i x \\ y &= i x \\ y &= \tan \left (\operatorname {RootOf}\left (-2 \textit {\_Z} \sqrt {a -1}-\ln \left (\sec \left (\textit {\_Z} \right )^{2} x^{2}\right )+2 c_{1} \right )\right ) x \\ y &= \tan \left (\operatorname {RootOf}\left (2 \textit {\_Z} \sqrt {a -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: 0.310 (sec). Leaf size: 125

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

\begin{align*} \text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {-K[1]^2+a-1}{\left (\sqrt {a-1}-K[1]\right ) \left (K[1]^2+1\right )}dK[1]&=-\log (x)+c_1,y(x)\right ] \\ \text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {-K[2]^2+a-1}{\left (K[2]+\sqrt {a-1}\right ) \left (K[2]^2+1\right )}dK[2]&=\log (x)+c_1,y(x)\right ] \\ y(x)\to -i x \\ y(x)\to i x \\ \end{align*}