2.35   ODE No. 35

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

$$f(x)(2ay(x)+b+y(x{)}^2)+y^{\prime }(x)=0$$ ✓ Mathematica : cpu = 0.0624036 (sec), leaf count = 48

$$\left\{\{y(x)\to \sqrt{b-a^2}\tan \left(\sqrt{b-a^2}({\int }_1^x-f(K[1])dK[1]+c_1)\right)-a\}\right\}$$

✓ Maple : cpu = 0.073 (sec), leaf count = 35

$$\{y\left(x\right)=\tanh \left(\sqrt{a^2-b}(\mathit{\_}\mathit{C}\mathit{1}+\int f\left(x\right)\mathrm{d}x)\right)\sqrt{a^2-b}-a\}$$

Hand solution

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

This is Riccati first order non-linear ODE. $P\left(x\right)=-bf\left(x\right),Q\left(x\right)=-2af\left(x\right),R\left(x\right)=-f\left(x\right)$.

Let $$y\left(x\right)=-\frac{u^{\prime }\left(x\right)}{u\left(x\right)R\left(x\right)}=\frac{u^{\prime }\left(x\right)}{u\left(x\right)f\left(x\right)}$$

Hence

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

Equating this to RHS of (1) gives

$$\begin{array}{rl}\frac{u^{\prime \prime }\left(x\right)}{u\left(x\right)f\left(x\right)}-\frac{{\left(u^{\prime }\left(x\right)\right)}^2}{u^2\left(x\right)f\left(x\right)}-\frac{u^{\prime }\left(x\right)f^{\prime }\left(x\right)}{u\left(x\right)f^2\left(x\right)}& =-2af\left(x\right)y\left(x\right)-bf\left(x\right)-f\left(x\right)y^2\left(x\right)\\ & =-2af\left(x\right)\left[\frac{u^{\prime }\left(x\right)}{u\left(x\right)f\left(x\right)}\right]-bf\left(x\right)-f\left(x\right){\left[\frac{u^{\prime }\left(x\right)}{u\left(x\right)f\left(x\right)}\right]}^2\\ & =-2a\frac{u^{\prime }\left(x\right)}{u\left(x\right)}-bf\left(x\right)-\frac{u^{\prime }{\left(x\right)}^2}{u^2\left(x\right)f\left(x\right)}\end{array}$$

Simplifying

$$\begin{array}{rl}{u}^{\prime \prime }\left(x\right)-\frac{{\left(u^{\prime }\left(x\right)\right)}^2}{u\left(x\right)}-\frac{u^{\prime }\left(x\right)f^{\prime }\left(x\right)}{f\left(x\right)}& =-2a{u}^{\prime }\left(x\right)f\left(x\right)-u\left(x\right)b{f}^2\left(x\right)-\frac{u^{\prime }{\left(x\right)}^2}{u\left(x\right)}\\ u^{\prime \prime }\left(x\right)-\frac{u^{\prime }\left(x\right)f^{\prime }\left(x\right)}{f\left(x\right)}& =-2a{u}^{\prime }\left(x\right)f\left(x\right)-u\left(x\right)b{f}^2\left(x\right)\\ u^{\prime \prime }\left(x\right)+u^{\prime }\left(x\right)(-\frac{f^{\prime }\left(x\right)}{f\left(x\right)}+2af\left(x\right))+u\left(x\right)b{f}^2\left(x\right)& =0\end{array}$$

Second order ODE with variable coefficients. Since coefficients are variables and not constants, a power series method is the standard way to continue. When I tried solving this now pretending the coefficients are constants in time, using the standard auxiliary equation method, the solution did verify OK. I need to look more into this. For now, this is solved using standard method for solving second order ODE with constant coefficients.

$$u\left(x\right)=C_1\exp \left(\frac{\int f\left(x\right)\sqrt{-b}dx(\sqrt{\frac{b-a^2}{b}}b+a\sqrt{-b})}{b}\right)+C_2\exp \left(\frac{\int f\left(x\right)\sqrt{-b}dx(-\sqrt{\frac{b-a^2}{b}}b+a\sqrt{-b})}{b}\right)$$

Hence

$$\begin{array}{rl}{u}^{\prime }\left(x\right)& =\frac{C_{1}f\left(x\right)\sqrt{-b}}{b}(\sqrt{\frac{-a^2+b}{b}}b+\sqrt{-b}a)e{}^{\frac{\int f\left(x\right)\sqrt{-b}\mathrm{d}x}{b}(\sqrt{\frac{-a^2+b}{b}}b+\sqrt{-b}a)}\\ & +\frac{C_{2}f\left(x\right)\sqrt{-b}}{b}(-\sqrt{\frac{-a^2+b}{b}}b+\sqrt{-b}a)e{}^{\frac{\int f\left(x\right)\sqrt{-b}\mathrm{d}x}{b}(-\sqrt{\frac{-a^2+b}{b}}b+\sqrt{-b}a)}\end{array}$$

Therefore

$$\begin{array}{rl}y& =\frac{u^{\prime }\left(x\right)}{u\left(x\right)f\left(x\right)}\\ & =\frac{\frac{C_{1}f\left(x\right)\sqrt{-b}}{b}(\sqrt{\frac{-a^2+b}{b}}b+\sqrt{-b}a)e{}^{\frac{\int f\left(x\right)\sqrt{-b}\mathrm{d}x}{b}(\sqrt{\frac{-a^2+b}{b}}b+\sqrt{-b}a)}+\frac{C_{2}f\left(x\right)\sqrt{-b}}{b}(-\sqrt{\frac{-a^2+b}{b}}b+\sqrt{-b}a)e{}^{\frac{\int f\left(x\right)\sqrt{-b}\mathrm{d}x}{b}(-\sqrt{\frac{-a^2+b}{b}}b+\sqrt{-b}a)}}{f\left(x\right)[C_1\exp \left(\frac{\int f\left(x\right)\sqrt{-b}dx(\sqrt{\frac{b-a^2}{b}}b+a\sqrt{-b})}{b}\right)+C_2\exp \left(\frac{\int f\left(x\right)\sqrt{-b}dx(-\sqrt{\frac{b-a^2}{b}}b+a\sqrt{-b})}{b}\right)]}\end{array}$$

Verification

restart; 
book:=diff(y(x),x)+f(x)*(2*a*y(x)+b+y(x)^2)=0; 
eqU:=diff(u(x),x$2)+diff(u(x),x)*(- diff(f(x),x)/f(x)+2*a*f(x))+u(x)*f(x)^2*b=0; 
solU:=dsolve(eqU,u(x)); 
my_sol:=diff(rhs(solU),x)/(rhs(solU)*f(x)); 
odetest(y(x)=my_sol,book); 
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