23.1.420 problem 409

Internal problem ID [5027]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 409
Date solved : Tuesday, September 30, 2025 at 11:27:26 AM
CAS classification : [_separable]

\begin{align*} y^{\prime } \sqrt {a_{4} x^{4}+a_{3} x^{3}+a_{2} x^{2}+a_{1} x +a_{0}}&=\sqrt {b_{0} +b_{1} y+b_{2} y^{2}} \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 76

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

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

✓ Mathematica. Time used: 70.47 (sec). Leaf size: 17788

ode=D[y[x],x]*Sqrt[a0+a1*x+a2*x^2+a3*x^3+a4*x^4]==Sqrt[b0+b1*y[x]+b2*y[x]^2]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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✓ Sympy. Time used: 14.733 (sec). Leaf size: 394

from sympy import * 
x = symbols("x") 
a0 = symbols("a0") 
a1 = symbols("a1") 
a2 = symbols("a2") 
a3 = symbols("a3") 
a4 = symbols("a4") 
b0 = symbols("b0") 
b1 = symbols("b1") 
b2 = symbols("b2") 
y = Function("y") 
ode = Eq(-sqrt(b0 + b1*y(x) + b2*y(x)**2) + sqrt(a0 + a1*x + a2*x**2 + a3*x**3 + a4*x**4)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = \begin {cases} C_{1} \sqrt {b_{0}} + \sqrt {b_{0}} \int \frac {1}{\sqrt {a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} + a_{4} x^{4}}}\, dx & \text {for}\: b_{1} = 0 \wedge b_{2} = 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} - \frac {b_{1}}{2 b_{2}} + e^{C_{1} \sqrt {b_{2}} + \sqrt {b_{2}} \int \frac {1}{\sqrt {a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} + a_{4} x^{4}}}\, dx} & \text {for}\: b_{0} - \frac {b_{1}^{2}}{4 b_{2}} = 0 \wedge b_{2} \neq 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} - b_{0} e^{- C_{1} \sqrt {b_{2}} - \sqrt {b_{2}} \int \frac {1}{\sqrt {a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} + a_{4} x^{4}}}\, dx} + \frac {b_{1}^{2} e^{- C_{1} \sqrt {b_{2}} - \sqrt {b_{2}} \int \frac {1}{\sqrt {a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} + a_{4} x^{4}}}\, dx}}{4 b_{2}} - \frac {b_{1}}{2 b_{2}} + \frac {e^{C_{1} \sqrt {b_{2}} + \sqrt {b_{2}} \int \frac {1}{\sqrt {a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} + a_{4} x^{4}}}\, dx}}{4 b_{2}} & \text {for}\: b_{2} \neq 0 \wedge b_{0} - \frac {b_{1}^{2}}{4 b_{2}} \neq 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} \frac {C_{1}^{2} b_{1}}{4} + \frac {C_{1} b_{1} \int \frac {1}{\sqrt {a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} + a_{4} x^{4}}}\, dx}{2} - \frac {b_{0}}{b_{1}} + \frac {b_{1} \left (\int \frac {1}{\sqrt {a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} + a_{4} x^{4}}}\, dx\right )^{2}}{4} & \text {for}\: b_{2} = 0 \wedge b_{1} \neq 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} - \frac {b_{1}}{2 b_{2}} + e^{C_{1} \sqrt {b_{2}} - \sqrt {b_{2}} \int \frac {1}{\sqrt {a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} + a_{4} x^{4}}}\, dx} & \text {for}\: b_{0} - \frac {b_{1}^{2}}{4 b_{2}} = 0 \wedge b_{2} \neq 0 \\\text {NaN} & \text {otherwise} \end {cases}\right ] \]