1.2 problem 1.1.2

Internal problem ID [10325]
Internal file name [OUTPUT/9273_Monday_June_06_2022_01_45_19_PM_3498373/index.tex]

Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section: Chapter 1, First-Order differential equations
Problem number: 1.1.2.
ODE order: 1.
ODE degree: 1.

The type(s) of ODE detected by this program : "quadrature"

Maple gives the following as the ode type

[_quadrature]

\[ \boxed {y^{\prime }-f \left (y\right )=0} \]

1.2.1 Solving as quadrature ode

Integrating both sides gives \begin {align*} \int _{}^{y}\frac {1}{f \left (\textit {\_a} \right )}d \textit {\_a} = x +c_{1} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{y}\frac {1}{f \left (\textit {\_a} \right )}d \textit {\_a} &= x +c_{1} \\ \end{align*}

Verification of solutions

\[ \int _{}^{y}\frac {1}{f \left (\textit {\_a} \right )}d \textit {\_a} = x +c_{1} \] Verified OK.

1.2.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-f \left (y\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=f \left (y\right ) \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{f \left (y\right )}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{f \left (y\right )}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Cannot compute integral}\hspace {3pt} \\ {} & {} & \int \frac {y^{\prime }}{f \left (y\right )}d x =x +c_{1} \end {array} \]

Maple trace

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

✓ Solution by Maple

Time used: 0.0 (sec). Leaf size: 17

dsolve(diff(y(x),x)=f(y(x)),y(x), singsol=all)
 

\[ x -\left (\int _{}^{y \left (x \right )}\frac {1}{f \left (\textit {\_a} \right )}d \textit {\_a} \right )+c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 0.442 (sec). Leaf size: 33

DSolve[y'[x]==f[y[x]],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{f(K[1])}dK[1]\&\right ][x+c_1] \\ y(x)\to f^{(-1)}(0) \\ \end{align*}