Internal problem ID [8730]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 394.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
\[ \boxed {{y^{\prime }}^{2}+2 f \left (x \right ) y y^{\prime }+g \left (x \right ) y^{2}=\left (g \left (x \right )-f \left (x \right )^{2}\right ) {\mathrm e}^{-2 \left (\int _{a}^{x}f \left (\operatorname {xp} \right )d \operatorname {xp} \right )}} \]
✓ Solution by Maple
Time used: 0.906 (sec). Leaf size: 107
dsolve(diff(y(x),x)^2+2*f(x)*y(x)*diff(y(x),x)+g(x)*y(x)^2-(g(x)-f(x)^2)*exp(-2*int(f(xp),xp = a .. x)) = 0,y(x), singsol=all)
\[ y \left (x \right ) = \tan \left (-\left (\int {\mathrm e}^{2 \left (\int _{a}^{x}f \left (\operatorname {xp} \right )d \operatorname {xp} \right )} \sqrt {-{\mathrm e}^{-4 \left (\int _{a}^{x}f \left (\operatorname {xp} \right )d \operatorname {xp} \right )} \left (f \left (x \right )^{2}-g \left (x \right )\right )}d x \right )+c_{1} \right ) \sqrt {{\cos \left (-\left (\int {\mathrm e}^{2 \left (\int _{a}^{x}f \left (\operatorname {xp} \right )d \operatorname {xp} \right )} \sqrt {-{\mathrm e}^{-4 \left (\int _{a}^{x}f \left (\operatorname {xp} \right )d \operatorname {xp} \right )} \left (f \left (x \right )^{2}-g \left (x \right )\right )}d x \right )+c_{1} \right )}^{2} {\mathrm e}^{-2 \left (\int _{a}^{x}f \left (\operatorname {xp} \right )d \operatorname {xp} \right )}} \]
✓ Solution by Mathematica
Time used: 60.339 (sec). Leaf size: 89
DSolve[-((-f[x]^2 + g[x])/E^(2*Integrate[f[xp], {xp, a, x}])) + g[x]*y[x]^2 + 2*f[x]*y[x]*y'[x] + y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-\int _a^x f(K[1]) \, dK[1]} \left ( \begin {array}{cc} \{ & \begin {array}{cc} \sin \left (c_1+\int _a^x \sqrt {g(K[1])-f(K[1])^2} \, dK[1]\right ) & g(x)>f(x)^2 \\ \cosh \left (c_1+\int _a^x \sqrt {f(K[1])^2-g(K[1])} \, dK[1]\right ) & g(x)<f(x)^2 \\ c_1 & \text {True} \\ \end {array} \\ \end {array} \right ) \]