Internal problem ID [7974]
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)]]]
Solve \begin {gather*} \boxed {\left (y^{\prime }\right )^{2}+2 f \relax (x ) y y^{\prime }+g \relax (x ) y^{2}-\left (g \relax (x )-f \relax (x )^{2}\right ) {\mathrm e}^{-2 \left (\int _{a}^{x}f \left (\mathit {xp} \right )d \mathit {xp} \right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 3.713 (sec). Leaf size: 131
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)
\[ y \relax (x ) = \tan \left (-\left (\int {\mathrm e}^{\int _{a}^{x}2 f \left (\mathit {xp} \right )d \mathit {xp}} \sqrt {-{\mathrm e}^{\int _{a}^{x}-4 f \left (\mathit {xp} \right )d \mathit {xp}} f \relax (x )^{2}+{\mathrm e}^{\int _{a}^{x}-4 f \left (\mathit {xp} \right )d \mathit {xp}} g \relax (x )}d x \right )+c_{1}\right ) \sqrt {\frac {{\mathrm e}^{-2 \left (\int _{a}^{x}f \left (\mathit {xp} \right )d \mathit {xp} \right )}}{\tan ^{2}\left (-\left (\int {\mathrm e}^{\int _{a}^{x}2 f \left (\mathit {xp} \right )d \mathit {xp}} \sqrt {-{\mathrm e}^{\int _{a}^{x}-4 f \left (\mathit {xp} \right )d \mathit {xp}} f \relax (x )^{2}+{\mathrm e}^{\int _{a}^{x}-4 f \left (\mathit {xp} \right )d \mathit {xp}} g \relax (x )}d x \right )+c_{1}\right )+1}} \]
✓ Solution by Mathematica
Time used: 59.558 (sec). Leaf size: 374
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]
\begin{align*} y(x)\to e^{-\int _a^x f(K[1]) \, dK[1]} \left ( {cc} \{ & {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} \\ \\ \\ \\ \right ) \\ y(x)\to \frac {i \sqrt {g(x)-f(x)^2} e^{-\int _a^x f(\text {xp}) \, d\text {xp}}}{\sqrt {f(x)^2-g(x)}} \\ y(x)\to \frac {i \sqrt {f(x)^2-g(x)} e^{-\int _a^x f(\text {xp}) \, d\text {xp}}}{\sqrt {g(x)-f(x)^2}} \\ y(x)\to {cc} \{ & {cc} \exp \left (-\text {Integrate}\left [f(K[1]),\{K[1],a,x\},\text {Assumptions}\to c_1\in \mathbb {R}\land 0<c_1<\frac {1}{4096}\land f(x)^2>g(x)\right ]\right ) \cosh \left (\text {Integrate}\left [\sqrt {f(K[1])^2-g(K[1])},\{K[1],a,x\},\text {Assumptions}\to c_1\in \mathbb {R}\land 0<c_1<\frac {1}{4096}\land f(x)^2>g(x)\right ]\right ) & f(x)^2-g(x)>0 \\ \exp \left (-\text {Integrate}\left [f(K[1]),\{K[1],a,x\},\text {Assumptions}\to c_1\in \mathbb {R}\land 0<c_1<\frac {1}{4096}\land f(x)^2<g(x)\right ]\right ) \sin \left (\text {Integrate}\left [\sqrt {g(K[1])-f(K[1])^2},\{K[1],a,x\},\text {Assumptions}\to c_1\in \mathbb {R}\land 0<c_1<\frac {1}{4096}\land f(x)^2<g(x)\right ]\right ) & f(x)^2-g(x)<0 \\ \\ \\ \\ \\ \end{align*}