Internal problem ID [532]
Internal file name [OUTPUT/532_Sunday_June_05_2022_01_43_10_AM_54801064/index.tex]
Book: Elementary differential equations and boundary value problems, 10th ed., Boyce and
DiPrima
Section: Section 2.5. Page 88
Problem number: 1.
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]
\[ \boxed {y^{\prime }-a y-b y^{2}=0} \]
Integrating both sides gives \begin {align*} \int \frac {1}{b \,y^{2}+a y}d y &= x +c_{1}\\ \frac {\ln \left (y \right )-\ln \left (b y +a \right )}{a}&=x +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=-\frac {{\mathrm e}^{a c_{1} +x a} a}{-1+b \,{\mathrm e}^{a c_{1} +x a}}\\ &=-\frac {c_{1} {\mathrm e}^{x a} a}{-1+b c_{1} {\mathrm e}^{x a}} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= -\frac {c_{1} {\mathrm e}^{x a} a}{-1+b c_{1} {\mathrm e}^{x a}} \\ \end{align*}
Verification of solutions
\[ y = -\frac {c_{1} {\mathrm e}^{x a} a}{-1+b c_{1} {\mathrm e}^{x a}} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-a y-b y^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=a y+b y^{2} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{a y+b y^{2}}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{a y+b y^{2}}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {\ln \left (y\right )}{a}-\frac {\ln \left (b y+a \right )}{a}=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=-\frac {{\mathrm e}^{c_{1} a +x a} a}{-1+b \,{\mathrm e}^{c_{1} a +x a}} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli <- Bernoulli successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 20
dsolve(diff(y(x),x) = a*y(x)+b*y(x)^2,y(x), singsol=all)
\[ y \left (x \right ) = \frac {a}{{\mathrm e}^{-a x} c_{1} a -b} \]
✓ Solution by Mathematica
Time used: 0.888 (sec). Leaf size: 45
DSolve[y'[x]== a*y[x]+b*y[x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {a e^{a (x+c_1)}}{-1+b e^{a (x+c_1)}} \\ y(x)\to 0 \\ y(x)\to -\frac {a}{b} \\ \end{align*}