60.1.72 problem 72

Internal problem ID [10086]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 72
Date solved : Monday, January 27, 2025 at 06:25:51 PM
CAS classification : [_separable]

\begin{align*} y^{\prime }-\operatorname {R1} \left (x , \sqrt {a_{4} x^{4}+a_{3} x^{3}+a_{2} x^{2}+a_{1} x +a_{0}}\right ) \operatorname {R2} \left (y, \sqrt {b_{4} y^{4}+b_{3} y^{3}+b_{2} y^{2}+b_{1} y+b_{0}}\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.003 (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)
 

\[ \int \operatorname {R1} \left (x , \sqrt {a_{4} x^{4}+a_{3} x^{3}+a_{2} x^{2}+a_{1} x +a_{0}}\right )d x -\int _{}^{y}\frac {1}{\operatorname {R2} \left (\textit {\_a} , \sqrt {\textit {\_a}^{4} b_{4} +\textit {\_a}^{3} b_{3} +\textit {\_a}^{2} b_{2} +\textit {\_a} b_{1} +b_{0}}\right )}d \textit {\_a} +c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 0.513 (sec). Leaf size: 86

DSolve[D[y[x],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]
 

\[ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{\text {R2}\left (K[1],\sqrt {\text {b4} K[1]^4+\text {b3} K[1]^3+\text {b2} K[1]^2+\text {b1} K[1]+\text {b0}}\right )}dK[1]\&\right ]\left [\int _1^x\text {R1}\left (K[2],\sqrt {\text {a0}+K[2] (\text {a1}+K[2] (\text {a2}+K[2] (\text {a3}+\text {a4} K[2])))}\right )dK[2]+c_1\right ] \]