23.3.241 problem 243

Internal problem ID [5955]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 243
Date solved : Tuesday, September 30, 2025 at 02:06:47 PM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} c y^{\prime }+\left (b x +a \right ) y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 27

ode:=c*diff(y(x),x)+(b*x+a)*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 +c_2 \left (\frac {b x +a}{b}\right )^{\frac {-c +b}{b}} \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 32

ode=c*D[y[x],x] + (a + b*x)*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {c_1 (a+b x)^{1-\frac {c}{b}}}{b-c}+c_2 \end{align*}

✓ Sympy. Time used: 0.838 (sec). Leaf size: 163

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
y = Function("y") 
ode = Eq(c*Derivative(y(x), x) + (a + b*x)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \begin {cases} - \frac {C_{1} a}{b e^{\frac {c \log {\left (a + b x \right )}}{b}} - c e^{\frac {c \log {\left (a + b x \right )}}{b}}} - \frac {C_{1} b x}{b e^{\frac {c \log {\left (a + b x \right )}}{b}} - c e^{\frac {c \log {\left (a + b x \right )}}{b}}} - \frac {C_{2} b e^{\frac {c \log {\left (a + b x \right )}}{b}}}{b e^{\frac {c \log {\left (a + b x \right )}}{b}} - c e^{\frac {c \log {\left (a + b x \right )}}{b}}} + \frac {C_{2} c e^{\frac {c \log {\left (a + b x \right )}}{b}}}{b e^{\frac {c \log {\left (a + b x \right )}}{b}} - c e^{\frac {c \log {\left (a + b x \right )}}{b}}} & \text {for}\: b \neq c \\- \frac {C_{1} \log {\left (\frac {a}{c} + x \right )}}{c} - C_{2} & \text {otherwise} \end {cases} \]