a):
(D2+1)y=3x−8cotx
Its auxiliary equation is m2+1=0
⇒m=±1
The complementary function is: yc(x)=c1cosx+c2sinx
Let u(x)=cosx,v(x)=sinx
Now, particular integral: yp(x)=A.u(x)+B.v(x)
We use the method of parameters.
A=−∫w(u,v)u(x)Q(x)dx,B=∫w(u,v)v(x)Q(X)dxw(u,v)=∣∣uu′vv′∣∣=∣∣cosx−sinxsinxcosx∣∣=1
And Q(x)=3x−8cotx
Then, A=−∫1cosx(3x−8cotx)dx
=−3∫xcosxdx+8∫cosxcotxdx=−3[xsinx−1×−cosx]+8[lntan(x/2)+cosx]=−3xsinx+5cosx+8lntan(x/2) [Using integration by parts]
Next, B=∫1sinx(3x−8cotx)dx
=3∫xsinxdx−8∫sinxcotxdx=3[xcosx−1×−sinx]−8sinx=−3xcosx−5sinx [Using integration by parts]
So, the particular integral is
yp(x)=A.u(x)+B.v(x)=(−3xsinx+5cosx+8lntan(x/2))(cosx)+(−3xcosx−5sinx)(sinx)=−6xsinxcosx+5[cos2x−sin2x]+8lntan(x/2)=−6xsinxcosx+5cos2x+8lntan(x/2)
Thus, the solution is
y=yc(x)+yp(x)y=c1cosx+c2sinx−6xsinxcosx+5cos2x+8lntan(x/2)
b):
Given curve is r=a(1+cosθ) ...(i)
Differentiating it, dr+asinθdθ=0 ...(ii)
On eliminating the parameter ' a ' we get,
dr+1+cosθrsinθdθ=0orr−1dθdr=1+cosθsinθ ...(iii)
This is the differential equation of the direction field D1 for the given family F1 .
To find the differential equation of the direction field orthological to D1, we replace dθdr
by −r2drdθ as evident from the geometry.
Where ϕ2 and Ψ2 are the angles for the orthogonal trajectory corresponding to the angles ϕ1 and Ψ1 for the given family (Fig. 1) as shown below.
Clearly, ϕ2(rdrdθ)2=(ϕ1±2π), whence tanϕ2=tan(ϕ1±2π)=−cotϕ1=−(r1dθdr)1
Whence from (3), the differential equation for the orthogonal trajectories becomes
rdrdθ=1+cosθsinθ=tan2θ
which is a case of variable-separable, and on integration gives
logr=2logsin2θ+log2c
This is the equation of the orthogonal family F2 .
Since r=c(1−cosθ) represents the same curve as r=c(1+cosθ), the member of
F2 with label c is the same as the member of F1 with label a=−c . Thus the given family is selforthogonal.
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