#### 2.34   ODE No. 34

$f(x) y(x)^2+g(x) y(x)+y'(x)=0$ Mathematica : cpu = 0.090378 (sec), leaf count = 54

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

$\left \{ y \left ( x \right ) ={\frac {{{\rm e}^{\int \!-g \left ( x \right ) \,{\rm d}x}}}{\int \!{{\rm e}^{\int \!-g \left ( x \right ) \,{\rm d}x}}f \left ( x \right ) \,{\rm d}x+{\it \_C1}}} \right \}$

Hand solution

\begin {align} y^{2}f+gy+y^{\prime } & =0\nonumber \\ y^{\prime } & =-gy-y^{2}f\nonumber \\ & =P\left ( x\right ) +Q\left ( x\right ) y+R\left ( x\right ) y^{2}\tag {1} \end {align}

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 {equation} \frac {y^{\prime }}{y^{2}}=-g\frac {1}{y}-f\tag {2} \end {equation}

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

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

Hence \begin {align*} y & =\frac {1}{e^{\int gdx}\left ( \int fe^{-\int gdx}+C\right ) }\\ & =\frac {e^{-\int gdx}}{\int fe^{-\int gdx}dx+C} \end {align*}

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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