1.1 problem 1.1.1

Internal problem ID [10324]
Internal file name [OUTPUT/9272_Monday_June_06_2022_01_45_19_PM_35985161/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: 1.1.1.
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]

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

1.1.1 Solving as quadrature ode

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

Summary

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

Verification of solutions

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

1.1.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }=f \left (x \right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int f \left (x \right )d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=\int f \left (x \right )d x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\int f \left (x \right )d x +c_{1} \end {array} \]

Maple trace

`Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
<- quadrature successful`
 

✓ Solution by Maple

Time used: 0.0 (sec). Leaf size: 10

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

\[ y \left (x \right ) = \int f \left (x \right )d x +c_{1} \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 18

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

\[ y(x)\to \int _1^xf(K[1])dK[1]+c_1 \]