2.1.18 Problem 18

Internal problem ID [8729]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 18
Date solved : Friday, April 25, 2025 at 04:58:17 PM
CAS classification : [_quadrature]

Existence and uniqueness analysis

Solve

$$\begin{array}{rl}{y}^{\prime }& =\sqrt{\frac{1+y}{y^2}}\end{array}$$

With initial conditions

$$\begin{array}{rl}y\left(0\right)& =1\end{array}$$

This is non linear first order ODE. In canonical form it is written as

$$\begin{array}{rl}{y}^{\prime }& =f(x,y)\\ & =\sqrt{\frac{1+y}{y^2}}\end{array}$$

The $y$ domain of $f(x,y)$ when $x=0$ is

$$\{-1\le y<0,0<y\le \mathrm{\infty }\}$$

And the point $y_0=1$ is inside this domain. Now we will look at the continuity of

$$\begin{array}{rl}\frac{\partial f}{\partial y}& =\frac{\partial }{\partial y}\left(\sqrt{\frac{1+y}{y^2}}\right)\\ & =\frac{\frac{1}{y^2}-\frac{2(1+y)}{y^3}}{2\sqrt{\frac{1+y}{y^2}}}\end{array}$$

The $y$ domain of $\frac{\partial f}{\partial y}$ when $x=0$ is

$$\{-\mathrm{\infty }\le y<-1,-1<y<0,0<y\le \mathrm{\infty }\}$$

And the point $y_0=1$ is inside this domain. Therefore solution exists and is unique.

Solved using first_order_ode_autonomous

Time used: 0.182 (sec)

Solve

$$\begin{array}{rl}{y}^{\prime }& =\sqrt{\frac{1+y}{y^2}}\end{array}$$

With initial conditions

$$\begin{array}{rl}y\left(0\right)& =1\end{array}$$

Integrating gives

$$\begin{array}{rl}\int \frac{1}{\sqrt{\frac{1+y}{y^2}}}dy& =dx\\ \frac{2\sqrt{\frac{1+y}{y^2}}y(y-2)}{3}& =x+c_1\end{array}$$

The following diagram is the phase line diagram. It classifies each of the above equilibrium points as stable or not stable or semi-stable.

Figure 2.41: Phase line diagram

Solving for the constant of integration from initial conditions, the solution becomes

$$\begin{array}{r}\frac{2\sqrt{\frac{1+y}{y^2}}y(y-2)}{3}=x-\frac{2\sqrt{2}}{3}\end{array}$$

Figure 2.42: Slope field $y^{\prime }=\sqrt{\frac{1+y}{y^2}}$

Summary of solutions found

$$\begin{array}{rl}\frac{2\sqrt{\frac{1+y}{y^2}}y(y-2)}{3}& =x-\frac{2\sqrt{2}}{3}\end{array}$$

✓ Maple. Time used: 0.381 (sec). Leaf size: 146

ode:=diff(y(x),x) = ((y(x)+1)/y(x)^2)^(1/2); 
ic:=y(0) = 1; 
dsolve([ode,ic],y(x), singsol=all);
 

$$y=-\frac{(1+i\sqrt{3}){(-12\sqrt{2}x+9x^2+\sqrt{(-12\sqrt{2}x+9x^2-8){(3x-2\sqrt{2})}^2})}^{2/3}-4i\sqrt{3}-4{(-12\sqrt{2}x+9x^2+\sqrt{(-12\sqrt{2}x+9x^2-8){(3x-2\sqrt{2})}^2})}^{1/3}+4}{4{(-12\sqrt{2}x+9x^2+\sqrt{(-12\sqrt{2}x+9x^2-8){(3x-2\sqrt{2})}^2})}^{1/3}}$$

Maple trace

Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
trying Bernoulli 
trying separable 
<- separable successful
 

Maple step by step

$$\begin{array}{lll}& & \text{Let’s solve}\\ & & [\frac{d}{dx}y\left(x\right)=\sqrt{\frac{1+y\left(x\right)}{y{\left(x\right)}^2}},y\left(0\right)=1]\\ \bullet & & \text{Highest derivative means the order of the ODE is}1\\ & & \frac{d}{dx}y\left(x\right)\\ \bullet & & \text{Solve for the highest derivative}\\ & & \frac{d}{dx}y\left(x\right)=\sqrt{\frac{1+y\left(x\right)}{y{\left(x\right)}^2}}\\ \bullet & & \text{Separate variables}\\ & & \frac{\frac{d}{dx}y\left(x\right)}{\sqrt{\frac{1+y\left(x\right)}{y{\left(x\right)}^2}}}=1\\ \bullet & & \text{Integrate both sides with respect to}x\\ & & \int \frac{\frac{d}{dx}y\left(x\right)}{\sqrt{\frac{1+y\left(x\right)}{y{\left(x\right)}^2}}}dx=\int 1dx+\mathit{C}\mathit{1}\\ \bullet & & \text{Evaluate integral}\\ & & \frac{2(1+y\left(x\right))(-2+y\left(x\right))}{3\sqrt{\frac{1+y\left(x\right)}{y{\left(x\right)}^2}}y\left(x\right)}=x+\mathit{C}\mathit{1}\\ \bullet & & \text{Solve for}y\left(x\right)\\ & & \{y\left(x\right)=\frac{{(-8+9{\mathit{C}\mathit{1}}^2+18\mathit{C}\mathit{1}x+9x^2+3\sqrt{9{\mathit{C}\mathit{1}}^4+36{\mathit{C}\mathit{1}}^{3}x+54{\mathit{C}\mathit{1}}^2{x}^2+36\mathit{C}\mathit{1}{x}^3+9x^4-16{\mathit{C}\mathit{1}}^2-32\mathit{C}\mathit{1}x-16x^2})}^{1/3}}{2}+\frac{2}{{(-8+9{\mathit{C}\mathit{1}}^2+18\mathit{C}\mathit{1}x+9x^2+3\sqrt{9{\mathit{C}\mathit{1}}^4+36{\mathit{C}\mathit{1}}^{3}x+54{\mathit{C}\mathit{1}}^2{x}^2+36\mathit{C}\mathit{1}{x}^3+9x^4-16{\mathit{C}\mathit{1}}^2-32\mathit{C}\mathit{1}x-16x^2})}^{1/3}}+1,y\left(x\right)=-\frac{{(-8+9{\mathit{C}\mathit{1}}^2+18\mathit{C}\mathit{1}x+9x^2+3\sqrt{9{\mathit{C}\mathit{1}}^4+36{\mathit{C}\mathit{1}}^{3}x+54{\mathit{C}\mathit{1}}^2{x}^2+36\mathit{C}\mathit{1}{x}^3+9x^4-16{\mathit{C}\mathit{1}}^2-32\mathit{C}\mathit{1}x-16x^2})}^{1/3}}{4}-\frac{1}{{(-8+9{\mathit{C}\mathit{1}}^2+18\mathit{C}\mathit{1}x+9x^2+3\sqrt{9{\mathit{C}\mathit{1}}^4+36{\mathit{C}\mathit{1}}^{3}x+54{\mathit{C}\mathit{1}}^2{x}^2+36\mathit{C}\mathit{1}{x}^3+9x^4-16{\mathit{C}\mathit{1}}^2-32\mathit{C}\mathit{1}x-16x^2})}^{1/3}}+1-\frac{\mathrm{I}\sqrt{3}(\frac{{(-8+9{\mathit{C}\mathit{1}}^2+18\mathit{C}\mathit{1}x+9x^2+3\sqrt{9{\mathit{C}\mathit{1}}^4+36{\mathit{C}\mathit{1}}^{3}x+54{\mathit{C}\mathit{1}}^2{x}^2+36\mathit{C}\mathit{1}{x}^3+9x^4-16{\mathit{C}\mathit{1}}^2-32\mathit{C}\mathit{1}x-16x^2})}^{1/3}}{2}-\frac{2}{{(-8+9{\mathit{C}\mathit{1}}^2+18\mathit{C}\mathit{1}x+9x^2+3\sqrt{9{\mathit{C}\mathit{1}}^4+36{\mathit{C}\mathit{1}}^{3}x+54{\mathit{C}\mathit{1}}^2{x}^2+36\mathit{C}\mathit{1}{x}^3+9x^4-16{\mathit{C}\mathit{1}}^2-32\mathit{C}\mathit{1}x-16x^2})}^{1/3}})}{2},y\left(x\right)=-\frac{{(-8+9{\mathit{C}\mathit{1}}^2+18\mathit{C}\mathit{1}x+9x^2+3\sqrt{9{\mathit{C}\mathit{1}}^4+36{\mathit{C}\mathit{1}}^{3}x+54{\mathit{C}\mathit{1}}^2{x}^2+36\mathit{C}\mathit{1}{x}^3+9x^4-16{\mathit{C}\mathit{1}}^2-32\mathit{C}\mathit{1}x-16x^2})}^{1/3}}{4}-\frac{1}{{(-8+9{\mathit{C}\mathit{1}}^2+18\mathit{C}\mathit{1}x+9x^2+3\sqrt{9{\mathit{C}\mathit{1}}^4+36{\mathit{C}\mathit{1}}^{3}x+54{\mathit{C}\mathit{1}}^2{x}^2+36\mathit{C}\mathit{1}{x}^3+9x^4-16{\mathit{C}\mathit{1}}^2-32\mathit{C}\mathit{1}x-16x^2})}^{1/3}}+1+\frac{\mathrm{I}\sqrt{3}(\frac{{(-8+9{\mathit{C}\mathit{1}}^2+18\mathit{C}\mathit{1}x+9x^2+3\sqrt{9{\mathit{C}\mathit{1}}^4+36{\mathit{C}\mathit{1}}^{3}x+54{\mathit{C}\mathit{1}}^2{x}^2+36\mathit{C}\mathit{1}{x}^3+9x^4-16{\mathit{C}\mathit{1}}^2-32\mathit{C}\mathit{1}x-16x^2})}^{1/3}}{2}-\frac{2}{{(-8+9{\mathit{C}\mathit{1}}^2+18\mathit{C}\mathit{1}x+9x^2+3\sqrt{9{\mathit{C}\mathit{1}}^4+36{\mathit{C}\mathit{1}}^{3}x+54{\mathit{C}\mathit{1}}^2{x}^2+36\mathit{C}\mathit{1}{x}^3+9x^4-16{\mathit{C}\mathit{1}}^2-32\mathit{C}\mathit{1}x-16x^2})}^{1/3}})}{2}\}\\ \bullet & & \text{Simplify}\\ & & \{y\left(x\right)=-\frac{(1+\mathrm{I}\sqrt{3}){(-8+9{\mathit{C}\mathit{1}}^2+18\mathit{C}\mathit{1}x+9x^2+3\sqrt{9}\sqrt{(\mathit{C}\mathit{1}+x+\frac{4}{3})(\mathit{C}\mathit{1}+x-\frac{4}{3}){(x+\mathit{C}\mathit{1})}^2})}^{2/3}-4\mathrm{I}\sqrt{3}-4{(-8+9{\mathit{C}\mathit{1}}^2+18\mathit{C}\mathit{1}x+9x^2+3\sqrt{9}\sqrt{(\mathit{C}\mathit{1}+x+\frac{4}{3})(\mathit{C}\mathit{1}+x-\frac{4}{3}){(x+\mathit{C}\mathit{1})}^2})}^{1/3}+4}{4{(-8+9{\mathit{C}\mathit{1}}^2+18\mathit{C}\mathit{1}x+9x^2+3\sqrt{9}\sqrt{(\mathit{C}\mathit{1}+x+\frac{4}{3})(\mathit{C}\mathit{1}+x-\frac{4}{3}){(x+\mathit{C}\mathit{1})}^2})}^{1/3}},y\left(x\right)=\frac{(\mathrm{I}\sqrt{3}-1){(-8+9{\mathit{C}\mathit{1}}^2+18\mathit{C}\mathit{1}x+9x^2+3\sqrt{9}\sqrt{(\mathit{C}\mathit{1}+x+\frac{4}{3})(\mathit{C}\mathit{1}+x-\frac{4}{3}){(x+\mathit{C}\mathit{1})}^2})}^{2/3}-4\mathrm{I}\sqrt{3}+4{(-8+9{\mathit{C}\mathit{1}}^2+18\mathit{C}\mathit{1}x+9x^2+3\sqrt{9}\sqrt{(\mathit{C}\mathit{1}+x+\frac{4}{3})(\mathit{C}\mathit{1}+x-\frac{4}{3}){(x+\mathit{C}\mathit{1})}^2})}^{1/3}-4}{4{(-8+9{\mathit{C}\mathit{1}}^2+18\mathit{C}\mathit{1}x+9x^2+3\sqrt{9}\sqrt{(\mathit{C}\mathit{1}+x+\frac{4}{3})(\mathit{C}\mathit{1}+x-\frac{4}{3}){(x+\mathit{C}\mathit{1})}^2})}^{1/3}},y\left(x\right)=\frac{{(-8+9{\mathit{C}\mathit{1}}^2+18\mathit{C}\mathit{1}x+9x^2+3\sqrt{(9{\mathit{C}\mathit{1}}^2+18\mathit{C}\mathit{1}x+9x^2-16){(x+\mathit{C}\mathit{1})}^2})}^{1/3}}{2}+\frac{2}{{(-8+9{\mathit{C}\mathit{1}}^2+18\mathit{C}\mathit{1}x+9x^2+3\sqrt{(9{\mathit{C}\mathit{1}}^2+18\mathit{C}\mathit{1}x+9x^2-16){(x+\mathit{C}\mathit{1})}^2})}^{1/3}}+1\}\\ \bullet & & \text{Use initial condition}y\left(0\right)=1\\ & & 1=-\frac{(1+\mathrm{I}\sqrt{3}){(-8+9{\mathit{C}\mathit{1}}^2+3\sqrt{9}\sqrt{(\mathit{C}\mathit{1}+\frac{4}{3})(\mathit{C}\mathit{1}-\frac{4}{3}){\mathit{C}\mathit{1}}^2})}^{2/3}-4\mathrm{I}\sqrt{3}-4{(-8+9{\mathit{C}\mathit{1}}^2+3\sqrt{9}\sqrt{(\mathit{C}\mathit{1}+\frac{4}{3})(\mathit{C}\mathit{1}-\frac{4}{3}){\mathit{C}\mathit{1}}^2})}^{1/3}+4}{4{(-8+9{\mathit{C}\mathit{1}}^2+3\sqrt{9}\sqrt{(\mathit{C}\mathit{1}+\frac{4}{3})(\mathit{C}\mathit{1}-\frac{4}{3}){\mathit{C}\mathit{1}}^2})}^{1/3}}\\ \bullet & & \text{Solve for}\mathit{\text{\_C1}}\\ & & \mathit{C}\mathit{1}=\mathit{RootOf}(\mathrm{I}\sqrt{3}{(-8+9{\mathit{\text{\_Z}}}^2+3\sqrt{9{\mathit{\text{\_Z}}}^4-16{\mathit{\text{\_Z}}}^2})}^{2/3}-4\mathrm{I}\sqrt{3}+{(-8+9{\mathit{\text{\_Z}}}^2+3\sqrt{9{\mathit{\text{\_Z}}}^4-16{\mathit{\text{\_Z}}}^2})}^{2/3}+4)\\ \bullet & & \mathrm{Remove}\mathrm{solutions}\mathrm{that}\mathrm{don}\mathrm{t}\mathrm{satisfy}\mathrm{the}\mathrm{ODE}\\ & & \varnothing \\ \bullet & & \text{Solution does not satisfy initial condition}\\ \bullet & & \text{Use initial condition}y\left(0\right)=1\\ & & 1=\frac{(\mathrm{I}\sqrt{3}-1){(-8+9{\mathit{C}\mathit{1}}^2+3\sqrt{9}\sqrt{(\mathit{C}\mathit{1}+\frac{4}{3})(\mathit{C}\mathit{1}-\frac{4}{3}){\mathit{C}\mathit{1}}^2})}^{2/3}-4\mathrm{I}\sqrt{3}+4{(-8+9{\mathit{C}\mathit{1}}^2+3\sqrt{9}\sqrt{(\mathit{C}\mathit{1}+\frac{4}{3})(\mathit{C}\mathit{1}-\frac{4}{3}){\mathit{C}\mathit{1}}^2})}^{1/3}-4}{4{(-8+9{\mathit{C}\mathit{1}}^2+3\sqrt{9}\sqrt{(\mathit{C}\mathit{1}+\frac{4}{3})(\mathit{C}\mathit{1}-\frac{4}{3}){\mathit{C}\mathit{1}}^2})}^{1/3}}\\ \bullet & & \text{Solve for}\mathit{\text{\_C1}}\\ & & \mathit{C}\mathit{1}=\mathit{RootOf}(\mathrm{I}\sqrt{3}{(-8+9{\mathit{\text{\_Z}}}^2+3\sqrt{9{\mathit{\text{\_Z}}}^4-16{\mathit{\text{\_Z}}}^2})}^{2/3}-{(-8+9{\mathit{\text{\_Z}}}^2+3\sqrt{9{\mathit{\text{\_Z}}}^4-16{\mathit{\text{\_Z}}}^2})}^{2/3}-4\mathrm{I}\sqrt{3}-4)\\ \bullet & & \mathrm{Remove}\mathrm{solutions}\mathrm{that}\mathrm{don}\mathrm{t}\mathrm{satisfy}\mathrm{the}\mathrm{ODE}\\ & & \varnothing \\ \bullet & & \text{Solution does not satisfy initial condition}\\ \bullet & & \text{Use initial condition}y\left(0\right)=1\\ & & 1=\frac{{(-8+9{\mathit{C}\mathit{1}}^2+3\sqrt{(9{\mathit{C}\mathit{1}}^2-16){\mathit{C}\mathit{1}}^2})}^{1/3}}{2}+\frac{2}{{(-8+9{\mathit{C}\mathit{1}}^2+3\sqrt{(9{\mathit{C}\mathit{1}}^2-16){\mathit{C}\mathit{1}}^2})}^{1/3}}+1\\ \bullet & & \text{Solve for}\mathit{\text{\_C1}}\\ & & \text{No solution}\\ \bullet & & \text{Solution does not satisfy initial condition}\end{array}$$

✓ Mathematica. Time used: 0.078 (sec). Leaf size: 123

ode=D[y[x],x]==Sqrt[ (1+y[x])/y[x]^2]; 
ic=y[0]==1; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

$$y(x)\to -\frac{1}{4}(1+i\sqrt{3})\sqrt[3]{9x^2+\sqrt{81x^4-216\sqrt{2}{x}^3+288x^2-64}-12\sqrt{2}x}+\frac{i(\sqrt{3}+i)}{\sqrt[3]{9x^2+\sqrt{81x^4-216\sqrt{2}{x}^3+288x^2-64}-12\sqrt{2}x}}+1$$

✓ Sympy. Time used: 0.366 (sec). Leaf size: 31

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-sqrt((y(x) + 1)/y(x)**2) + Derivative(y(x), x),0) 
ics = {y(0): 1} 
dsolve(ode,func=y(x),ics=ics)
 

$$\stackrel{y\left(x\right)}{\int }\frac{1}{\sqrt{\frac{1+\frac{1}{y}}{y}}}dy=x+\stackrel{1}{\int }\frac{1}{\sqrt{\frac{1}{y}+\frac{1}{y^2}}}dy$$