Internal problem ID [13309]
Internal file name [OUTPUT/12481_Wednesday_February_14_2024_02_06_40_AM_38606342/index.tex]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 4. SEPARABLE FIRST ORDER EQUATIONS. Additional exercises. page
90
Problem number: 4.4 (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]
\[ \boxed {y^{\prime }-y^{2}=9} \]
Integrating both sides gives \begin {align*} \int \frac {1}{y^{2}+9}d y &= x +c_{1}\\ \frac {\arctan \left (\frac {y}{3}\right )}{3}&=x +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=3 \tan \left (3 c_{1} +3 x \right ) \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= 3 \tan \left (3 c_{1} +3 x \right ) \\ \end{align*}
Verification of solutions
\[ y = 3 \tan \left (3 c_{1} +3 x \right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-y^{2}=9 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=y^{2}+9 \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{y^{2}+9}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{y^{2}+9}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {\arctan \left (\frac {y}{3}\right )}{3}=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=3 \tan \left (3 c_{1} +3 x \right ) \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable <- separable successful`
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 14
dsolve(diff(y(x),x)=y(x)^2+9,y(x), singsol=all)
\[ y \left (x \right ) = 3 \tan \left (3 c_{1} +3 x \right ) \]
✓ Solution by Mathematica
Time used: 0.199 (sec). Leaf size: 28
DSolve[y'[x]==y[x]^2+9,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to 3 \tan (3 (x+c_1)) \\ y(x)\to -3 i \\ y(x)\to 3 i \\ \end{align*}