2.94   ODE No. 94

1. Problem in Latex
2. Mathematica input
3. Maple input

$$ay(x)+b{x}^n+x{y}^{\prime }(x)=0$$ ✓ Mathematica : cpu = 0.0173539 (sec), leaf count = 25

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

$$\{y\left(x\right)=-\frac{b{x}^n}{n+a}+x^{-a}\mathit{\_}\mathit{C}\mathit{1}\}$$

Hand solution

$$x{y}^{\prime }+ay+b{x}^n=0$$

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

If $a=-n$ then

$$\begin{array}{rl}{x}^{a}y& =-\int b{x}^{-1}+C\\ y& =-x^{-a}b\ln \left(x\right)+x^{-a}C\\ & =x^{-a}(C-b\ln x)\end{array}$$

If $a\ne -n$ then

$$\begin{array}{rl}{x}^{a}y& =-\frac{b{x}^{a+n}}{a+n}+C\\ y& =-b\frac{x^n}{a+n}+C{x}^{-a}\end{array}$$

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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