Internal problem ID [14070]
Internal file name [OUTPUT/13751_Saturday_March_02_2024_02_49_16_PM_43060135/index.tex]
Book: INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton.
Fourth edition 2014. ElScAe. 2014
Section: Chapter 1. Introduction to Differential Equations. Exercises 1.1, page 10
Problem number: 34.
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]
\[ \boxed {y^{\prime }=\left (x^{2}-1\right ) \left (x^{3}-3 x \right )^{3}} \]
Integrating both sides gives \begin {align*} y &= \int { \left (x^{2}-1\right ) x^{3} \left (x^{2}-3\right )^{3}\,\mathop {\mathrm {d}x}}\\ &= \frac {x^{4} \left (x^{8}-12 x^{6}+54 x^{4}-108 x^{2}+81\right )}{12}+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {x^{4} \left (x^{8}-12 x^{6}+54 x^{4}-108 x^{2}+81\right )}{12}+c_{1} \\ \end{align*}
Verification of solutions
\[ y = \frac {x^{4} \left (x^{8}-12 x^{6}+54 x^{4}-108 x^{2}+81\right )}{12}+c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }=\left (x^{2}-1\right ) \left (x^{3}-3 x \right )^{3} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \left (x^{2}-1\right ) \left (x^{3}-3 x \right )^{3}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=\frac {\left (x^{3}-3 x \right )^{4}}{12}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {1}{12} x^{12}-x^{10}+\frac {9}{2} x^{8}-9 x^{6}+\frac {27}{4} x^{4}+c_{1} \end {array} \]
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: 31
dsolve(diff(y(x),x)=(x^2-1)*(x^3-3*x)^3,y(x), singsol=all)
\[ y \left (x \right ) = \frac {1}{12} x^{12}-x^{10}+\frac {9}{2} x^{8}-9 x^{6}+\frac {27}{4} x^{4}+c_{1} \]
✓ Solution by Mathematica
Time used: 0.002 (sec). Leaf size: 39
DSolve[y'[x]==(x^2-1)*(x^3-3*x)^3,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {x^{12}}{12}-x^{10}+\frac {9 x^8}{2}-9 x^6+\frac {27 x^4}{4}+c_1 \]