Internal problem ID [3003]
Internal file name [OUTPUT/2495_Sunday_June_05_2022_03_16_47_AM_24614121/index.tex]
Book: Theory and solutions of Ordinary Differential equations, Donald Greenspan,
1960
Section: Exercises, page 14
Problem number: 1(b).
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]
\[ \boxed {y^{\prime }=1-x^{5}+\sqrt {x}} \]
Integrating both sides gives \begin {align*} y &= \int { 1-x^{5}+\sqrt {x}\,\mathop {\mathrm {d}x}}\\ &= -\frac {x^{6}}{6}+\frac {2 x^{\frac {3}{2}}}{3}+x +c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {x^{6}}{6}+\frac {2 x^{\frac {3}{2}}}{3}+x +c_{1} \\ \end{align*}
Verification of solutions
\[ y = -\frac {x^{6}}{6}+\frac {2 x^{\frac {3}{2}}}{3}+x +c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }=1-x^{5}+\sqrt {x} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \left (1-x^{5}+\sqrt {x}\right )d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=-\frac {x^{6}}{6}+\frac {2 x^{\frac {3}{2}}}{3}+x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-\frac {x^{6}}{6}+\frac {2 x^{\frac {3}{2}}}{3}+x +c_{1} \end {array} \]
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: 17
dsolve(diff(y(x),x)=1-x^5+sqrt(x),y(x), singsol=all)
\[ y \left (x \right ) = \frac {2 x^{\frac {3}{2}}}{3}-\frac {x^{6}}{6}+x +c_{1} \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 25
DSolve[y'[x]==1-x^5+Sqrt[x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {2 x^{3/2}}{3}-\frac {x^6}{6}+x+c_1 \]