Internal problem ID [13453]
Internal file name [OUTPUT/12625_Friday_February_16_2024_12_00_55_AM_90671311/index.tex]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 8. Review exercises for part of part II. page 143
Problem number: 32.
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]
\[ \boxed {y^{\prime } \left (x +2\right )=x^{3}} \]
Integrating both sides gives \begin {align*} y &= \int { \frac {x^{3}}{x +2}\,\mathop {\mathrm {d}x}}\\ &= \frac {x^{3}}{3}-x^{2}+4 x -8 \ln \left (x +2\right )+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {x^{3}}{3}-x^{2}+4 x -8 \ln \left (x +2\right )+c_{1} \\ \end{align*}
Verification of solutions
\[ y = \frac {x^{3}}{3}-x^{2}+4 x -8 \ln \left (x +2\right )+c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime } \left (x +2\right )=x^{3} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {x^{3}}{x +2} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \frac {x^{3}}{x +2}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=\frac {x^{3}}{3}-x^{2}+4 x -8 \ln \left (x +2\right )+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {x^{3}}{3}-x^{2}+4 x -8 \ln \left (x +2\right )+c_{1} \end {array} \]
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: 25
dsolve((x+2)*diff(y(x),x)-x^3=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {x^{3}}{3}-x^{2}+4 x -8 \ln \left (x +2\right )+c_{1} \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 32
DSolve[(x+2)*y'[x]-x^3==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {x^3}{3}-x^2+4 x-8 \log (x+2)+\frac {44}{3}+c_1 \]