Internal problem ID [14079]
Internal file name [OUTPUT/13760_Saturday_March_02_2024_02_49_18_PM_26374251/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: 43.
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]
\[ \boxed {y^{\prime }=\left (-x^{2}+4\right )^{\frac {3}{2}}} \]
Integrating both sides gives \begin {align*} y &= \int { \left (-x^{2}+4\right )^{\frac {3}{2}}\,\mathop {\mathrm {d}x}}\\ &= \frac {x \left (x^{2}-10\right ) \left (x^{2}-4\right )}{4 \sqrt {-x^{2}+4}}+6 \arcsin \left (\frac {x}{2}\right )+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {x \left (x^{2}-10\right ) \left (x^{2}-4\right )}{4 \sqrt {-x^{2}+4}}+6 \arcsin \left (\frac {x}{2}\right )+c_{1} \\ \end{align*}
Verification of solutions
\[ y = \frac {x \left (x^{2}-10\right ) \left (x^{2}-4\right )}{4 \sqrt {-x^{2}+4}}+6 \arcsin \left (\frac {x}{2}\right )+c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }=\left (-x^{2}+4\right )^{\frac {3}{2}} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \left (-x^{2}+4\right )^{\frac {3}{2}}d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=\frac {x \left (-x^{2}+4\right )^{\frac {3}{2}}}{4}+\frac {3 \sqrt {-x^{2}+4}\, x}{2}+6 \arcsin \left (\frac {x}{2}\right )+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {x \left (-x^{2}+4\right )^{\frac {3}{2}}}{4}+\frac {3 \sqrt {-x^{2}+4}\, x}{2}+6 \arcsin \left (\frac {x}{2}\right )+c_{1} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful`
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 32
dsolve(diff(y(x),x)=(4-x^2)^(3/2),y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (-x^{3}+10 x \right ) \sqrt {-x^{2}+4}}{4}+c_{1} +6 \arcsin \left (\frac {x}{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.062 (sec). Leaf size: 49
DSolve[y'[x]==(4-x^2)^(3/2),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -12 \arctan \left (\frac {\sqrt {4-x^2}}{x+2}\right )-\frac {1}{4} x \sqrt {4-x^2} \left (x^2-10\right )+c_1 \]