Internal problem ID [13260]
Internal file name [OUTPUT/12432_Wednesday_February_14_2024_02_06_12_AM_29904459/index.tex]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 2. Integration and differential equations. Additional exercises. page
32
Problem number: 2.3 (i).
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]
\[ \boxed {9 y^{\prime }=x^{2}-1} \]
Integrating both sides gives \begin {align*} y &= \int { \frac {x^{2}}{9}-\frac {1}{9}\,\mathop {\mathrm {d}x}}\\ &= \frac {x \left (x^{2}-3\right )}{27}+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {x \left (x^{2}-3\right )}{27}+c_{1} \\ \end{align*}
Verification of solutions
\[ y = \frac {x \left (x^{2}-3\right )}{27}+c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 9 y^{\prime }=x^{2}-1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {x^{2}}{9}-\frac {1}{9} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \left (\frac {x^{2}}{9}-\frac {1}{9}\right )d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=\frac {1}{27} x^{3}-\frac {1}{9} x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {1}{27} x^{3}-\frac {1}{9} x +c_{1} \end {array} \]
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: 14
dsolve(1=x^2-9*diff(y(x),x),y(x), singsol=all)
\[ y \left (x \right ) = \frac {1}{27} x^{3}-\frac {1}{9} x +c_{1} \]
✓ Solution by Mathematica
Time used: 0.002 (sec). Leaf size: 20
DSolve[1==x^2-9*y'[x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {x^3}{27}-\frac {x}{9}+c_1 \]