2.5E: Exercises for Section 2.5

Exercise \(\PageIndex{1}\)

Find the family of orthogonal curves to the given family of curves: \( y^2 = a x^3 \).

Answer

\( \frac {x^2}{3} + \frac {y^2}{2}=C, \text { This represents family of ellipses.}\)

Exercise \(\PageIndex{2}\)

Find the family of orthogonal curves to the given family of curves: \( y = a e^x \).

Answer

\(y^2+2x=C\)

Exercise \(\PageIndex{3}\)

Find the family of orthogonal curves to the given family of curves: \(y^2 = a x^4\).

Answer

\(\frac{x^2}{2} + y^2 = C\)

Exercise \(\PageIndex{4}\)

Find the family of orthogonal curves to the given family of curves: \( y = a e^{2x} \).

Answer

\(y^2+x=C\)

Exercise \(\PageIndex{5}\)

Find the family of orthogonal curves to the given family of curves: \( y = a x^2 \).

Answer

\(\frac{x^2}{2} + y^2=C\)

Exercise \(\PageIndex{6}\)

Find the family of orthogonal curves to the family of coaxial circles: \( x^2 + y^2+2gx +c = 0 \), where g is a parameter.

Hint: Differentiate the equation and find g and substitute in the given equation and find differential equation independent of g, before you proceed.

Answer

 \( x^2 = - y^2 +c + ky; \text { c and k are constants of integration. You can write it as } x^2 + y^2 +2fy - c =0 : \text { where} k = -2f. \)

Exercise \(\PageIndex{7}\)

Find the family of orthogonal curves to the given family of curves: \(x^2+4xy+y^2=c\)

Answer

\((y-x)^3(y+x)=k\) or \((x-y)^3(x+y)=k_1 \)

Exercise \(\PageIndex{8}\)

Find the family of orthogonal curves to the given family of curves: \(y=ce^2x\)

Answer

\(y^2 + x =C\)

Exercise \(\PageIndex{9}\)

Find the family of orthogonal curves to the given family of curves: \(xye^{x^2}=c\)

Answer

\(y^2=-\frac{1}{2}\ln(1+2x^2)+k\)

Exercise \(\PageIndex{10}\)

Find the family of orthogonal curves to the given family of curves: \(y=\frac{ce^x}{x}\)

Answer

\(y^2 + 2x +2Ln \lvert x-1\rvert =C\)

Exercise \(\PageIndex{11}\)

A cat sitting in a field suddenly sees a standing dog. To save its life, the cat runs away in a straight line with speed \( u \). Without any delay, the dog starts running with constant speed \( v > u \) to catch the cat. Initially, \( v \) is perpendicular to \( u \) and \( L \) is the initial separation between the two. If the dog always changes its direction so that it is always heading directly at the cat, find the time the dog takes to catch the cat in terms of \( v, u, \) and \( L \). Assume that:

• the dog starts at the point \(S = (L,0)\) and the cat at the origin \(O = (0,0)\).
• the cat moves in the positive direction along the \(y\)-axis, and the dog's path describes a curve of pursuit.

Answer

\(T = L \dfrac{v}{v^2 - u^2}. \) 

Exercise \(\PageIndex{12}\)

Show the system of confocal conics \( \frac{x^2}{a^2+ \lambda } + \frac{y^2}{b^2+ \lambda } =1 \) is self orthogonal.

Answer

Hint: Differentiate w.r.t 'x' and find \( \lambda = - \frac{b^2x +a^2py}{x+py} \text { where } p = \frac{dy}{dx}. \) 

Solve to write differential equation of the given conic is \( (x+py)( x-\frac{y}{p})= a^2-b^2 \text { after replacing p with} -\frac{1}{p} \text { we get the same equation } ( x-\frac{y}{p})(x+py)= a^2-b^2.\) It shows syestem is self orthogonal.