Internal problem ID [3381]
Internal file name [OUTPUT/2873_Sunday_June_05_2022_08_43_32_AM_71396820/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 5
Problem number: 123.
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]
\[ \boxed {y^{\prime }-b \sin \left (y\right )=a} \]
Integrating both sides gives \begin {align*} \int \frac {1}{a +b \sin \left (y \right )}d y &= x +c_{1}\\ \frac {2 \arctan \left (\frac {2 a \tan \left (\frac {y}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}}&=x +c_{1} \end {align*}
Solving for \(y\) gives these solutions \begin {align*} y_1&=2 \arctan \left (\frac {\tan \left (\frac {c_{1} \sqrt {a^{2}-b^{2}}}{2}+\frac {x \sqrt {a^{2}-b^{2}}}{2}\right ) \sqrt {a^{2}-b^{2}}-b}{a}\right ) \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= 2 \arctan \left (\frac {\tan \left (\frac {c_{1} \sqrt {a^{2}-b^{2}}}{2}+\frac {x \sqrt {a^{2}-b^{2}}}{2}\right ) \sqrt {a^{2}-b^{2}}-b}{a}\right ) \\ \end{align*}
Verification of solutions
\[ y = 2 \arctan \left (\frac {\tan \left (\frac {c_{1} \sqrt {a^{2}-b^{2}}}{2}+\frac {x \sqrt {a^{2}-b^{2}}}{2}\right ) \sqrt {a^{2}-b^{2}}-b}{a}\right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-b \sin \left (y\right )=a \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=a +b \sin \left (y\right ) \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{a +b \sin \left (y\right )}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{a +b \sin \left (y\right )}d x =\int 1d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {2 \arctan \left (\frac {2 \tan \left (\frac {y}{2}\right ) a +2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}}=x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=2 \arctan \left (\frac {\tan \left (\frac {c_{1} \sqrt {a^{2}-b^{2}}}{2}+\frac {x \sqrt {a^{2}-b^{2}}}{2}\right ) \sqrt {a^{2}-b^{2}}-b}{a}\right ) \end {array} \]
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: 44
dsolve(diff(y(x),x) = a+b*sin(y(x)),y(x), singsol=all)
\[ y \left (x \right ) = 2 \arctan \left (\frac {-b +\tan \left (\frac {\sqrt {a^{2}-b^{2}}\, \left (c_{1} +x \right )}{2}\right ) \sqrt {a^{2}-b^{2}}}{a}\right ) \]
✓ Solution by Mathematica
Time used: 60.173 (sec). Leaf size: 52
DSolve[y'[x]==a+b Sin[y[x]],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to 2 \arctan \left (\frac {-b+\sqrt {a^2-b^2} \tan \left (\frac {1}{2} \sqrt {a^2-b^2} (x+c_1)\right )}{a}\right ) \]