2.26   ODE No. 26

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

$$y^{\prime }(x)-(Ay(x)-a)(By(x)-b)=0$$ ✓ Mathematica : cpu = 0.293557 (sec), leaf count = 56

$$\left\{\{y(x)\to \frac{a{e}^{Ab(c_1+x)}-b{e}^{aB(c_1+x)}}{A{e}^{Ab(c_1+x)}-B{e}^{aB(c_1+x)}}\}\right\}$$

✓ Maple : cpu = 0.1 (sec), leaf count = 45

$$\{y\left(x\right)=\frac{{\mathrm{e}}^{(x+\mathit{\_}\mathit{C}\mathit{1})(Ab-aB)}a-b}{A{\mathrm{e}}^{(x+\mathit{\_}\mathit{C}\mathit{1})(Ab-aB)}-B}\}$$

Hand solution

$$\begin{array}{rl}{y}^{\prime }-(Ay-a)(By-b)& =0\\ y^{\prime }& =(Ay-a)(By-b)\\ \text{(1)}& & =ab-y(Ab+Ba)+AB{y}^2\end{array}$$

This is Riccati first order non-linear ODE with $P\left(x\right)=ab,Q\left(x\right)=-(Ab+Ba),R\left(x\right)=AB$. Let $y=-\frac{u^{\prime }}{uR\left(x\right)}=-\frac{u^{\prime }}{ABu}$, hence

$$y^{\prime }=\frac{-u^{\prime \prime }}{ABu}-\frac{{\left(u^{\prime }\right)}^2}{AB{u}^2}$$

Comparing to (1) results in

$$\begin{array}{rl}\frac{-u^{\prime \prime }}{ABu}-\frac{{\left(u^{\prime }\right)}^2}{AB{u}^2}& =ab-y(Ab+Ba)+AB{y}^2\\ & =ab-(-\frac{u^{\prime }}{ABu})(Ab+Ba)+AB{(-\frac{u^{\prime }}{ABu})}^2\\ & =ab+\frac{u^{\prime }}{ABu}(Ab+Ba)+AB\frac{{\left(u^{\prime }\right)}^2}{{\left(ABu\right)}^2}\\ & =ab+\frac{u^{\prime }}{ABu}(Ab+Ba)+\frac{{\left(u^{\prime }\right)}^2}{AB{u}^2}\end{array}$$

Hence

$$\begin{array}{rl}\frac{-u^{\prime \prime }}{ABu}& =ab+\frac{u^{\prime }}{ABu}(Ab+Ba)\\ -u^{\prime \prime }& =ABabu+u^{\prime }(Ab+Ba)\\ u^{\prime \prime }+u^{\prime }(Ab+Ba)+u\left(ABab\right)& =0\end{array}$$

This is second order ODE with constant coefficient. Solution is

$$u=c_1{e}^{-aBx}+c_2{e}^{-Abx}$$

Therefore

$$u^{\prime }=-aB{c}_1{e}^{-aBx}-c_{2}Ab{e}^{-Abx}$$

And therefore the solution is

$$\begin{array}{rl}y& =-\frac{u^{\prime }}{ABu}=-\frac{1}{AB}\frac{-aB{c}_1{e}^{-aBx}-c_{2}Ab{e}^{-Abx}}{c_1{e}^{-aBx}+c_2{e}^{-Abx}}\\ & =\frac{aB{c}_1{e}^{-aBx}+c_{2}Ab{e}^{-Abx}}{AB(c_1{e}^{-aBx}+c_2{e}^{-Abx})}\end{array}$$

Dividing by $c_2$ and letting $c=\frac{c_1}{c_2}$

$$y=\frac{aBc{e}^{-aBx}+Ab{e}^{-Abx}}{AB(c{e}^{-aBx}+e^{-Abx})}$$

Verification

eq:=diff(y(x),x)-(A*y(x)-a)*(B*y(x)-b) = 0; 
sol:=(a*B*_C1*exp(-a*B*x)+A*b*exp(-A*b*x))/(A*B*(_C1*exp(-a*B*x)+exp(-A*b*x))); 
odetest(y(x)=sol,eq); 
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