2.1004   ODE No. 1004

\[ -a \cos (b x)+y''(x)+y(x)=0 \]

✓ Mathematica : cpu = 0.0076557 (sec), leaf count = 30

DSolve[-(a*Cos[b*x]) + y[x] + Derivative[2][y][x] == 0,y[x],x]
 

\[\left \{\left \{y(x)\to -\frac {a \cos (b x)}{b^2-1}+c_1 \cos (x)+c_2 \sin (x)\right \}\right \}\]

✓ Maple : cpu = 0.227 (sec), leaf count = 27

dsolve(diff(diff(y(x),x),x)+y(x)-a*cos(b*x)=0,y(x))
 

\[y \left (x \right ) = \sin \left (x \right ) c_{2} +\cos \left (x \right ) c_{1} -\frac {a \cos \left (b x \right )}{b^{2}-1}\]

Hand solution

\begin{equation} y^{\prime \prime }+y=a\cos bx \tag {1}\end{equation}

We start by solving the homogeneous equation

\[ y^{\prime \prime }+y=0 \]

Let \(y=e^{\lambda x}\), substitution in above gives

\begin{align*} \lambda ^{2}e^{\lambda x}+e^{\lambda x} & =0\\ \lambda ^{2}+1 & =0 \end{align*}

Hence \(\lambda =\pm i\), therefore the solution is

\begin{align*} y_{h} & =Ae^{ix}+Be^{-ix}\\ & =A\left ( \cos x+i\sin x\right ) +B\left ( \cos x-i\sin x\right ) \\ & =\cos x\left ( A+B\right ) +\sin x\left ( Ai-iB\right ) \\ & =\cos x\left ( A+B\right ) +\sin x\left ( i\left ( A-B\right ) \right ) \end{align*}

Let \(A+B=c_{1},i\left ( A-B\right ) =c_{2}\) hence

\[ y_{h}=c_{1}\cos x+c_{2}\sin x \]

Now we solve for the particular solution using variation of parameters. Let

\begin{align*} y_{p} & =u_{1}\left ( x\right ) \cos x+u_{2}\left ( x\right ) \sin x\\ y_{p}^{\prime } & =u_{1}^{\prime }\cos x-u_{1}\sin x+u_{2}^{\prime }\sin x+u_{2}\cos x\\ & =u_{2}\cos x-u_{1}\sin x+u_{1}^{\prime }\cos x+u_{2}^{\prime }\sin x \end{align*}

Let first condition be

\begin{equation} u_{1}^{\prime }\cos x+u_{2}^{\prime }\sin x=0 \tag {2}\end{equation}

Hence

\begin{align*} y_{p}^{\prime } & =u_{2}\cos x-u_{1}\sin x\\ y_{p}^{\prime \prime } & =u_{2}^{\prime }\cos x-u_{2}\sin x-u_{1}^{\prime }\sin x-u_{1}\cos x \end{align*}

Substituting in (1) gives

\begin{align} y_{p}^{\prime \prime }+y_{p} & =a\cos bx\nonumber \\ u_{2}^{\prime }\cos x-u_{2}\sin x-u_{1}^{\prime }\sin x-u_{1}\cos x+u_{1}\cos x+u_{2}\sin x & =a\cos bx\nonumber \\ u_{2}^{\prime }\cos x-u_{1}^{\prime }\sin x & =a\cos bx\tag {3}\end{align}

So we have two equations (1)(2) to solve for \(u_{1},u_{2}\)

\begin{align*} u_{1}^{\prime }\cos x+u_{2}^{\prime }\sin x & =0\\ u_{2}^{\prime }\cos x-u_{1}^{\prime }\sin x & =a\cos bx \end{align*}

From the first equation

\begin{equation} u_{1}^{\prime }=-u_{2}^{\prime }\frac {\sin x}{\cos x}\tag {4}\end{equation}

Substituting in the second equation

\begin{align*} u_{2}^{\prime }\cos x-\left ( -u_{2}^{\prime }\frac {\sin x}{\cos x}\right ) \sin x & =a\cos bx\\ u_{2}^{\prime }\left ( \cos x+\frac {\sin x}{\cos x}\sin x\right ) & =a\cos bx\\ u_{2}^{\prime }\left ( \frac {\cos ^{2}x+\sin ^{2}x}{\cos x}\right ) & =a\cos bx\\ u_{2}^{\prime } & =a\cos x\cos bx \end{align*}

Hence

\begin{align*} u_{2} & =a\int \cos x\cos \left ( bx\right ) dx\\ & =a\frac {-\cos \left ( bx\right ) \sin x+b\cos x\sin \left ( bx\right ) }{b^{2}-1}\end{align*}

From (4)

\begin{align*} u_{1}^{\prime } & =-a\cos \left ( bx\right ) \sin x\\ u_{1} & =-a\int \cos \left ( bx\right ) \sin xdx\\ & =-a\frac {\cos \left ( bx\right ) \cos x+b\sin x\sin \left ( bx\right ) }{b^{2}-1}\end{align*}

Since \(y_{p}=u_{1}\left ( x\right ) \cos x+u_{2}\left ( x\right ) \sin x\) then

\begin{align*} y_{p} & =\left ( -a\frac {\cos \left ( bx\right ) \cos x+b\sin x\sin \left ( bx\right ) }{b^{2}-1}\right ) \cos x+\left ( a\frac {-\cos \left ( bx\right ) \sin x+b\cos x\sin \left ( bx\right ) }{b^{2}-1}\right ) \sin x\\ & =\frac {-a\cos \left ( bx\right ) \cos ^{2}x-ab\cos x\sin x\sin \left ( bx\right ) -a\cos \left ( bx\right ) \sin ^{2}x+ab\sin x\cos x\sin \left ( bx\right ) }{b^{2}-1}\\ & =\frac {-a\cos \left ( bx\right ) \cos ^{2}x-a\cos \left ( bx\right ) \sin ^{2}x}{b^{2}-1}\\ & =\frac {-a\cos \left ( bx\right ) \left ( \cos ^{2}x+\sin ^{2}x\right ) }{b^{2}-1}\\ & =\frac {-a\cos \left ( bx\right ) }{b^{2}-1}\\ & =\frac {a\cos \left ( bx\right ) }{1-b^{2}}\end{align*}

Therefore, the full solution is (for \(b^{2}\neq 1\))

\begin{align*} y & =y_{h}+y_{p}\\ & =c_{1}\cos x+c_{2}\sin x+\frac {a\cos \left ( bx\right ) }{1-b^{2}}\end{align*}

restart; 
ode:=diff(diff(y(x),x),x)+y(x)-a*cos(b*x)=0; 
y0:=_C1*cos(x)+_C2*sin(x)+a*cos(b*x)/(1-b^2); 
odetest(y(x)=y0,ode); 
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