Question #283546

a) Solve (𝐷


2 + 1)𝑦 = 3𝑥 − 8 cot 𝑥 where 𝐷 =


𝑑


𝑑𝑥



b) Find orthogonal trajectory to the curve given by 𝑟 = 𝑎(1 + cos 𝜃)

1
Expert's answer
2022-01-04T07:29:40-0500

Solution:

a):

(D2+1)y=3x8cotx(D^2+1)y=3x-8\cot x

Its auxiliary equation is m2+1=0m^2+1=0

m=±1\Rightarrow m=\pm1

The complementary function is: yc(x)=c1cosx+c2sinxy_c(x)=c_1\cos x+c_2\sin x

Let u(x)=cosx,v(x)=sinxu(x)=\cos x,v(x)=\sin x

Now, particular integral: yp(x)=A.u(x)+B.v(x)y_p(x)=A.u(x)+B.v(x)

We use the method of parameters.

A=u(x)Q(x)w(u,v)dx,B=v(x)Q(X)w(u,v)dxw(u,v)=uvuv=cosxsinxsinxcosx=1A=-\int \dfrac{u(x)Q(x)}{w(u,v)}dx,B=\int \dfrac{v(x)Q(X)}{w(u,v)}dx \\w(u,v)=\begin{vmatrix} u&v\\u'&v'\end{vmatrix}=\begin{vmatrix} \cos x&\sin x\\-\sin x&\cos x\end{vmatrix}=1

And Q(x)=3x8cotxQ(x)=3x-8\cot x

Then, A=cosx(3x8cotx)1dxA=-\int \dfrac{\cos x(3x-8\cot x)}{1}dx

=3xcosxdx+8cosxcotxdx=3[xsinx1×cosx]+8[lntan(x/2)+cosx]=3xsinx+5cosx+8lntan(x/2)=-3\int x\cos xdx+8\int\cos x\cot xdx \\=-3[x \sin x-1\times -\cos x]+8[\ln \tan(x/2)+\cos x] \\=-3x\sin x+5\cos x+8 \ln \tan(x/2) [Using integration by parts]

Next, B=sinx(3x8cotx)1dxB=\int \dfrac{\sin x(3x-8\cot x)}{1}dx

=3xsinxdx8sinxcotxdx=3[xcosx1×sinx]8sinx=3xcosx5sinx=3\int x\sin xdx-8\int\sin x\cot xdx \\=3[x\cos x-1\times-\sin x]-8\sin x \\=-3x\cos x-5\sin x [Using integration by parts]

So, the particular integral is

yp(x)=A.u(x)+B.v(x)=(3xsinx+5cosx+8lntan(x/2))(cosx)+(3xcosx5sinx)(sinx)=6xsinxcosx+5[cos2xsin2x]+8lntan(x/2)=6xsinxcosx+5cos2x+8lntan(x/2)y_p(x)=A.u(x)+B.v(x) \\=(-3x\sin x+5\cos x+8 \ln \tan(x/2))(\cos x)+(-3x\cos x-5\sin x)(\sin x) \\=-6x\sin x \cos x+5[\cos^2x-\sin^2x]+8\ln \tan(x/2) \\=-6x\sin x \cos x+5\cos2x+8\ln \tan(x/2)

Thus, the solution is

y=yc(x)+yp(x)y=c1cosx+c2sinx6xsinxcosx+5cos2x+8lntan(x/2)y=y_c(x)+y_p(x) \\y=c_1\cos x+c_2\sin x-6x\sin x \cos x+5\cos2x+8\ln \tan(x/2)

b):

Given curve is r=a(1+cosθ)r=a(1+\cos \theta) ...(i)

Differentiating it, dr+asinθdθ=0d r+a \sin \theta d \theta=0 ...(ii)

On eliminating the parameter ' a ' we get,

dr+rsinθ1+cosθdθ=0or1rdrdθ=sinθ1+cosθd r+\frac{r \sin \theta}{1+\cos \theta} d \theta=0 \quad or \quad \frac{-1}{r} \frac{d r}{d \theta}=\frac{\sin \theta}{1+\cos \theta} ...(iii)

This is the differential equation of the direction field D1D_{1} for the given family F1F_{1} .

To find the differential equation of the direction field orthological to D1D_{1}, we replace drdθ\frac{d r}{d \theta}

by r2dθdr-r^{2} \frac{d \theta}{d r} as evident from the geometry.

Where ϕ2 and Ψ2\phi_{2}\ and\ \Psi_{2} are the angles for the orthogonal trajectory corresponding to the angles ϕ1 and Ψ1\phi_{1}\ and\ \Psi_{1} for the given family (Fig. 1) as shown below.


 Clearly, ϕ2=(ϕ1±π2), whence tanϕ2=tan(ϕ1±π2)=cotϕ1(rdθdr)2=(1rdrdθ)1\begin{aligned} \text { Clearly, } \quad \phi_{2} &=\left(\phi_{1} \pm \frac{\pi}{2}\right) \text {, whence } \tan \phi_{2}=\tan \left(\phi_{1} \pm \frac{\pi}{2}\right)=-\cot \phi_{1} \\ \left(r \frac{d \theta}{d r}\right)_{2} &=-\left(\frac{1}{r} \frac{d r}{d \theta}\right)_{1} \end{aligned}

Whence from (3), the differential equation for the orthogonal trajectories becomes

rdθdr=sinθ1+cosθ=tanθ2r \frac{d \theta}{d r}=\frac{\sin \theta}{1+\cos \theta}=\tan \frac{\theta}{2}

which is a case of variable-separable, and on integration gives

logr=2logsinθ2+log2c\log r=2 \log \sin \frac{\theta}{2}+\log 2 c

This is the equation of the orthogonal family F2F_{2} .

Since r=c(1cosθ)r=c(1-\cos \theta) represents the same curve as r=c(1+cosθ),r=c(1+\cos \theta), the member of

F2F_{2} with label c is the same as the member of F1F_{1} with label a=ca=-c . Thus the given family is selforthogonal.


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