2.77   ODE No. 77

\[ y'(x)-\cos (a y(x)+b x)=0 \] Mathematica : cpu = 0.313988 (sec), leaf count = 124


\[\left \{\left \{y(x)\to \frac {-b x-2 \tan ^{-1}\left (\frac {a \tanh \left (\frac {1}{2} \left (-x \sqrt {a^2-b^2}+c_1 \sqrt {a^2-b^2}\right )\right )}{\sqrt {a^2-b^2}}+\frac {b \tanh \left (\frac {1}{2} \left (-x \sqrt {a^2-b^2}+c_1 \sqrt {a^2-b^2}\right )\right )}{\sqrt {a^2-b^2}}\right )}{a}\right \}\right \}\] Maple : cpu = 0.08 (sec), leaf count = 54


\[y \relax (x ) = \frac {-b x +2 \arctan \left (\frac {\tanh \left (\frac {\sqrt {\left (a -b \right ) \left (a +b \right )}\, \left (x -c_{1}\right )}{2}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}{a -b}\right )}{a}\]

Hand solution

\[ y^{\prime }=\cos \left (ay+bx\right ) \]

This is separable after transformation of \(u=ay+bx\), hence \(u^{\prime }=ay^{\prime }+b\) or \(y^{\prime }=\frac {1}{a}\left (u^{\prime }-b\right ) \). Therefore the above becomes

\begin {align*} \frac {1}{a}\left (u^{\prime }-b\right ) & =\cos \relax (u) \\ u^{\prime } & =a\cos u+b\\ \frac {du}{a\cos u+b} & =dx \end {align*}

This is the same as Kamke 76 (the problem before this), which we solved using half angle tan transformation, and the answer is

\[ u=2\arctan \left (\frac {a+b}{\sqrt {b^{2}-a^{2}}}\tan \left (\frac {1}{2}\sqrt {b^{2}-a^{2}}\left (x+C\right ) \right ) \right ) \]

Since \(u=ay+bx\) then \(y=\frac {u-bx}{a}\), hence

\[ y=\frac {1}{a}\left (2\arctan \left (\frac {a+b}{\sqrt {b^{2}-a^{2}}}\tan \left ( \frac {1}{2}\sqrt {b^{2}-a^{2}}\left (x+C\right ) \right ) \right ) -bx\right ) \]

Verification