Internal problem ID [7653]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 72.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_separable]
Solve \begin {gather*} \boxed {y^{\prime }-\mathit {R1} \left (x , \sqrt {a_{4} x^{4}+a_{3} x^{3}+a_{2} x^{2}+a_{1} x +a_{0}}\right ) \mathit {R2} \left (y, \sqrt {b_{4} y^{4}+b_{3} y^{3}+b_{2} y^{2}+b_{1} y+b_{0}}\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 64
dsolve(diff(y(x),x) - R1(x,sqrt(a__4*x^4+a__3*x^3+a__2*x^2+a__1*x+a__0))*R2(y(x),sqrt(b__4*y(x)^4+b__3*y(x)^3+b__2*y(x)^2+b__1*y(x)+b__0))=0,y(x), singsol=all)
\[ \int \mathit {R1} \left (x , \sqrt {a_{4} x^{4}+a_{3} x^{3}+a_{2} x^{2}+a_{1} x +a_{0}}\right )d x -\left (\int _{}^{y \relax (x )}\frac {1}{\mathit {R2} \left (\textit {\_a} , \sqrt {\textit {\_a}^{4} b_{4} +\textit {\_a}^{3} b_{3} +\textit {\_a}^{2} b_{2} +\textit {\_a} b_{1} +b_{0}}\right )}d \textit {\_a} \right )+c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.683 (sec). Leaf size: 86
DSolve[y'[x] - R1[x,Sqrt[a4*x^4+a3*x^3+a2*x^2+a1*x+a0]]*R2[y[x],Sqrt[b4*y[x]^4+b3*y[x]^3+b2*y[x]^2+b1*y[x]+b0]]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{\text {R2}\left (K[1],\sqrt {\text {b4} K[1]^4+\text {b3} K[1]^3+\text {b2} K[1]^2+\text {b1} K[1]+\text {b0}}\right )}dK[1]\&\right ]\left [\int _1^x\text {R1}\left (K[2],\sqrt {\text {a0}+K[2] (\text {a1}+K[2] (\text {a2}+K[2] (\text {a3}+\text {a4} K[2])))}\right )dK[2]+c_1\right ] \\ \end{align*}