2.34   ODE No. 34

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

$$f(x)y(x{)}^2+g(x)y(x)+y^{\prime }(x)=0$$ ✓ Mathematica : cpu = 0.090378 (sec), leaf count = 54

$$\left\{\{y(x)\to \frac{\exp ({\int }_1^x-g(K[1])dK[1])}{-{\int }_1^x-\exp ({\int }_1^{K[2]}-g(K[1])dK[1])f(K[2])dK[2]+c_1}\}\right\}$$ ✓ Maple : cpu = 0.024 (sec), leaf count = 28

$$\{y\left(x\right)=\frac{{\mathrm{e}}^{\int -g\left(x\right)\mathrm{d}x}}{\int {\mathrm{e}}^{\int -g\left(x\right)\mathrm{d}x}f\left(x\right)\mathrm{d}x+\mathit{\_}\mathit{C}\mathit{1}}\}$$

Hand solution

$$\begin{array}{rl}{y}^{2}f+gy+y^{\prime }& =0\\ y^{\prime }& =-gy-y^{2}f\\ \text{(1)}& & =P\left(x\right)+Q\left(x\right)y+R\left(x\right)y^2\end{array}$$

This is Bernoulli first order non-linear ODE. $P\left(x\right)=0,Q\left(x\right)=-g,R\left(x\right)=f$. First step is to divide by $y^2$$$\begin{array}{}\text{(2)}& \frac{y^{\prime }}{y^2}=-g\frac{1}{y}-f\end{array}$$

Let $u=\frac{1}{y}$, then $u^{\prime }=\frac{-y^{\prime }}{y^2}$ and (2) becomes$$\begin{array}{rl}-u^{\prime }& =-gu-f\\ u^{\prime }-gu& =f\end{array}$$

Integrating factor is $e^{-\int gdx}$ hence$$\begin{array}{rl}d\left(e^{-\int gdx}u\right)& =f{e}^{-\int gdx}\\ e^{-\int gdx}u& =\int f{e}^{-\int gdx}dx+C\\ u& =e^{\int gdx}(\int f{e}^{-\int gdx}dx+C)\end{array}$$

Hence $$\begin{array}{rl}y& =\frac{1}{e^{\int gdx}(\int f{e}^{-\int gdx}+C)}\\ & =\frac{e^{-\int gdx}}{\int f{e}^{-\int gdx}dx+C}\end{array}$$

Let $\beta =e^{-\int gdx}$ then$$y=\frac{\beta }{\int f\beta dx+C}$$

Verification

restart; 
eq:=diff(y(x),x)+f(x)*y(x)^2+g(x)*y(x) = 0; 
beta:=exp(-Int(g(x),x)): 
my_sol:=beta/(Int(f(x)*beta,x)+_C1); 
odetest(y(x)=my_sol,eq); 
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