Internal problem ID [13274]
Internal file name [OUTPUT/12446_Wednesday_February_14_2024_02_06_18_AM_50862479/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: 2.6 (a).
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]
\[ \boxed {y^{\prime }=3 \sqrt {3+x}} \]
Integrating both sides gives \begin {align*} y &= \int { 3 \sqrt {3+x}\,\mathop {\mathrm {d}x}}\\ &= 2 \left (3+x \right )^{\frac {3}{2}}+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= 2 \left (3+x \right )^{\frac {3}{2}}+c_{1} \\ \end{align*}
Verification of solutions
\[ y = 2 \left (3+x \right )^{\frac {3}{2}}+c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }=3 \sqrt {3+x} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int 3 \sqrt {3+x}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=2 \left (3+x \right )^{\frac {3}{2}}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=2 \left (3+x \right )^{\frac {3}{2}}+c_{1} \end {array} \]
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)=3*sqrt(x+3),y(x), singsol=all)
\[ y \left (x \right ) = \left (2 x +6\right ) \sqrt {x +3}+c_{1} \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 17
DSolve[y'[x]==3*Sqrt[x+3],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to 2 (x+3)^{3/2}+c_1 \]