Internal problem ID [1146]
Internal file name [OUTPUT/1147_Sunday_June_05_2022_02_03_32_AM_85019389/index.tex]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 5 linear second order equations. Section 5.6 Reduction or order. Page
253
Problem number: 38 part (e).
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]
\[ \boxed {y^{\prime }+y^{2}+14 y=-50} \]
Integrating both sides gives \begin {align*} \int \frac {1}{-y^{2}-14 y -50}d y &= x +c_{1}\\ -\arctan \left (7+y \right )&=x +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=-7-\tan \left (x +c_{1} \right ) \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -7-\tan \left (x +c_{1} \right ) \\ \end{align*}
Verification of solutions
\[ y = -7-\tan \left (x +c_{1} \right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+y^{2}+14 y=-50 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-y^{2}-14 y-50 \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{-y^{2}-14 y-50}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{-y^{2}-14 y-50}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\arctan \left (7+y\right )=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-7-\tan \left (x +c_{1} \right ) \end {array} \]
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: 0.016 (sec). Leaf size: 12
dsolve(diff(y(x),x)+y(x)^2+14*y(x)+50=0,y(x), singsol=all)
\[ y \left (x \right ) = -7-\tan \left (c_{1} +x \right ) \]
✓ Solution by Mathematica
Time used: 0.553 (sec). Leaf size: 30
DSolve[y'[x]+y[x]^2+14*y[x]+50==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -7-\tan (x-c_1) \\ y(x)\to -7-i \\ y(x)\to -7+i \\ \end{align*}