1.24 problem 24

Internal problem ID [5737]
Internal file name [OUTPUT/4985_Sunday_June_05_2022_03_15_57_PM_66201785/index.tex]

Book: Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold Scientific. Singapore. 1995
Section: Chapter 1. First order differential equations. Section 1.1 Separable equations problems. page 7
Problem number: 24.
ODE order: 1.
ODE degree: 1.

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

Maple gives the following as the ode type

[_separable]

\[ \boxed {z^{\prime }-10^{x +z}=0} \]

1.24.1 Solving as separable ode

In canonical form the ODE is \begin {align*} z' &= F(x,z)\\ &= f( x) g(z)\\ &= 10^{x} 10^{z} \end {align*}

Where \(f(x)=10^{x}\) and \(g(z)=10^{z}\). Integrating both sides gives \begin{align*} \frac {1}{10^{z}} \,dz &= 10^{x} \,d x \\ \int { \frac {1}{10^{z}} \,dz} &= \int {10^{x} \,d x} \\ -\frac {10^{-z}}{\ln \left (10\right )}&=\frac {10^{x}}{\ln \left (10\right )}+c_{1} \\ \end{align*} The solution is \[ -\frac {10^{-z}}{\ln \left (10\right )}-\frac {10^{x}}{\ln \left (10\right )}-c_{1} = 0 \]

Summary

The solution(s) found are the following \begin{align*} \tag{1} -\frac {10^{-z}}{\ln \left (10\right )}-\frac {10^{x}}{\ln \left (10\right )}-c_{1} &= 0 \\ \end{align*}

Figure 19: Slope field plot

Verification of solutions

\[ -\frac {10^{-z}}{\ln \left (10\right )}-\frac {10^{x}}{\ln \left (10\right )}-c_{1} = 0 \] Verified OK.

1.24.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & z^{\prime }-10^{x +z}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & z^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & z^{\prime }=10^{x +z} \end {array} \]

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: 0.015 (sec). Leaf size: 29

dsolve(diff(z(x),x)=10^(x+z(x)),z(x), singsol=all)
 

\[ z \left (x \right ) = \frac {\ln \left (-\frac {1}{c_{1} \ln \left (2\right )+c_{1} \ln \left (5\right )+10^{x}}\right )}{\ln \left (2\right )+\ln \left (5\right )} \]

✓ Solution by Mathematica

Time used: 0.93 (sec). Leaf size: 24

DSolve[z'[x]==10^(x+z[x]),z[x],x,IncludeSingularSolutions -> True]
 

\[ z(x)\to -\frac {\log \left (-10^x+c_1 (-\log (10))\right )}{\log (10)} \]