60.1.71 problem 72

Internal problem ID [10085]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 72
Date solved : Sunday, March 30, 2025 at 03:13:39 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*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 64

ode:=diff(y(x),x)-R1(x,(a__4*x^4+a__3*x^3+a__2*x^2+a__1*x+a__0)^(1/2))*R2(y(x),(b__4*y(x)^4+b__3*y(x)^3+b__2*y(x)^2+b__1*y(x)+b__0)^(1/2)) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ \int \operatorname {R1} \left (x , \sqrt {a_{4} x^{4}+a_{3} x^{3}+a_{2} x^{2}+a_{1} x +a_{0}}\right )d x -\int _{}^{y}\frac {1}{\operatorname {R2} \left (\textit {\_a} , \sqrt {\textit {\_a}^{4} b_{4} +\textit {\_a}^{3} b_{3} +\textit {\_a}^{2} b_{2} +\textit {\_a} b_{1} +b_{0}}\right )}d \textit {\_a} +c_1 = 0 \]

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

ode=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; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 1.182 (sec). Leaf size: 58

from sympy import * 
x = symbols("x") 
a__0 = symbols("a__0") 
a__1 = symbols("a__1") 
a__2 = symbols("a__2") 
a__3 = symbols("a__3") 
a__4 = symbols("a__4") 
b__0 = symbols("b__0") 
b__1 = symbols("b__1") 
b__2 = symbols("b__2") 
b__3 = symbols("b__3") 
b__4 = symbols("b__4") 
y = Function("y") 
ode = Eq(-R1(x, sqrt(a__0 + a__1*x + a__2*x**2 + a__3*x**3 + a__4*x**4))*R2(y(x), sqrt(b__0 + b__1*y(x) + b__2*y(x)**2 + b__3*y(x)**3 + b__4*y(x)**4)) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \int \limits ^{y{\left (x \right )}} \frac {1}{R_{2}{\left (y,\sqrt {y^{4} b^{4} + y^{3} b^{3} + y^{2} b^{2} + y b^{1} + b^{0}} \right )}}\, dy = C_{1} + \int R_{1}{\left (x,\sqrt {a^{0} + a^{1} x + a^{2} x^{2} + a^{3} x^{3} + a^{4} x^{4}} \right )}\, dx \]