Internal problem ID [13426]
Internal file name [OUTPUT/12598_Thursday_February_15_2024_12_01_01_AM_47669130/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: 5.
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]
\[ \boxed {x^{2} y^{\prime }=\sqrt {x}+3} \]
Integrating both sides gives \begin {align*} y &= \int { \frac {\sqrt {x}+3}{x^{2}}\,\mathop {\mathrm {d}x}}\\ &= -\frac {2}{\sqrt {x}}-\frac {3}{x}+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {2}{\sqrt {x}}-\frac {3}{x}+c_{1} \\ \end{align*}
Verification of solutions
\[ y = -\frac {2}{\sqrt {x}}-\frac {3}{x}+c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{2} y^{\prime }=\sqrt {x}+3 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\sqrt {x}+3}{x^{2}} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \frac {\sqrt {x}+3}{x^{2}}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=-\frac {2}{\sqrt {x}}-\frac {3}{x}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {c_{1} x^{\frac {3}{2}}-3 \sqrt {x}-2 x}{x^{\frac {3}{2}}} \end {array} \]
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: 16
dsolve(x^2*diff(y(x),x)-sqrt(x)=3,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {3}{x}-\frac {2}{\sqrt {x}}+c_{1} \]
✓ Solution by Mathematica
Time used: 0.011 (sec). Leaf size: 20
DSolve[x^2*y'[x]-Sqrt[x]==3,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -\frac {2}{\sqrt {x}}-\frac {3}{x}+c_1 \]