2.12 problem 17

Internal problem ID [14116]
Internal file name [OUTPUT/13797_Saturday_March_02_2024_02_50_43_PM_14517269/index.tex]

Book: INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section: Chapter 1. Introduction to Differential Equations. Review exercises, page 23
Problem number: 17.
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]

\[ \boxed {y^{\prime }=\sin \left (x \right ) x^{2}} \]

2.12.1 Solving as quadrature ode

Integrating both sides gives \begin {align*} y &= \int { \sin \left (x \right ) x^{2}\,\mathop {\mathrm {d}x}}\\ &= \left (-x^{2}+2\right ) \cos \left (x \right )+2 \sin \left (x \right ) x +c_{1} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= \left (-x^{2}+2\right ) \cos \left (x \right )+2 \sin \left (x \right ) x +c_{1} \\ \end{align*}

Figure 105: Slope field plot

Verification of solutions

\[ y = \left (-x^{2}+2\right ) \cos \left (x \right )+2 \sin \left (x \right ) x +c_{1} \] Verified OK.

2.12.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }=\sin \left (x \right ) x^{2} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \sin \left (x \right ) x^{2}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=-\cos \left (x \right ) x^{2}+2 \cos \left (x \right )+2 \sin \left (x \right ) x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-\cos \left (x \right ) x^{2}+2 \cos \left (x \right )+2 \sin \left (x \right ) x +c_{1} \end {array} \]

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: 22

dsolve(diff(y(x),x)=x^2*sin(x),y(x), singsol=all)
 

\[ y \left (x \right ) = -x^{2} \cos \left (x \right )+2 \cos \left (x \right )+2 x \sin \left (x \right )+c_{1} \]

✓ Solution by Mathematica

Time used: 0.018 (sec). Leaf size: 22

DSolve[y'[x]==x^2*Sin[x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to -\left (x^2-2\right ) \cos (x)+2 x \sin (x)+c_1 \]