#### 2.94   ODE No. 94

$a y(x)+b x^n+x y'(x)=0$ Mathematica : cpu = 0.0173539 (sec), leaf count = 25

$\left \{\left \{y(x)\to -\frac {b x^n}{a+n}+c_1 x^{-a}\right \}\right \}$ Maple : cpu = 0.009 (sec), leaf count = 23

$\left \{ y \left ( x \right ) =-{\frac {b{x}^{n}}{n+a}}+{x}^{-a}{\it \_C1} \right \}$

Hand solution

$xy^{\prime }+ay+bx^{n}=0$

Linear ﬁrst order, exact, separable. $$y^{\prime }+\frac {ay}{x}=-bx^{n-1}$$, integrating factor $$\mu =e^{\int \frac {a}{x}dx}=e^{a\ln x}=x^{a}$$, hence\begin {align*} d\left ( \mu y\right ) & =-\mu bx^{n-1}\\ x^{a}y & =-\int bx^{a+n-1}+C \end {align*}

If $$a=-n$$ then

\begin {align*} x^{a}y & =-\int bx^{-1}+C\\ y & =-x^{-a}b\ln \left ( x\right ) +x^{-a}C\\ & =x^{-a}\left ( C-b\ln x\right ) \end {align*}

If $$a\neq -n$$ then

\begin {align*} x^{a}y & =-\frac {bx^{a+n}}{a+n}+C\\ y & =-b\frac {x^{n}}{a+n}+Cx^{-a} \end {align*}

Veriﬁcation

restart;
ode:=x*diff(y(x),x)+a*y(x)+b*x^n=0;
s1:=x^(-a)*(_C1-b*ln(x));
s2:=-b*(x^n/(a+n))+_C1*x^(-a);
odetest(y(x)=s2,ode);
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