OP Malhotra Class 12 Maths Solutions Chapter 17 Differential Equations Ex 17(b)

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S Chand Class 12 ICSE Maths Solutions Chapter 17 Differential Equations Ex 17(b)

Question 1.
Write the differential equation representing the family of curves y = nix, where m is an arbitrary constant.
Solution:
Given family of curves be,
y = mx …(1)
where m be an arbitrary constant.
Diff. both sides of eqn. (1) w.r.t. x; we have
\(\frac { dy }{ dx }\) = m … (2)
putting the valueof m from (2) in eqn. (1) ; we have
y = x \(\frac { dy }{ dx }\) be the required diff. eqn.

Question 2.
Form the differential equation by eliminating the parameters A and B from the equation y =Ae^ax + Be^-ax.
Solution:
Given eqn. be,
y = Ae^ax + Be^-ax …(1)
Diff. eqn. (1) both sides w.r.t. x, we have
\(\frac { dy }{ dx }\) = Ae^ax – Bae^-ax …(2)
Diff. eqn. (2) both sides w.r.t. x, we have
\(\frac{d^2 y}{d x^2}=\mathrm{A} a^2 e^{a x}+\mathrm{B} a^2 e^{-a x}\) = a²(Ae^ax + Be^-ax)
⇒ \(\frac{d^2 y}{d x^2}\) [using eqn. (1)]

Question 3.
Form the differntial equation represen-ting the family of curves
y = tan^-1 x + c etan^-1 x, where c is an arbitrary constant.
Solution:
Given eqn. of family of curves be
y = tan^-1 x + c etan^-1 x … (1)
where c be any arbitrary constant diff. both sides w.r.t. x; we have

which is the required diff. eqn.

Question 4.
Form the differential equation representing the family of curves: y = e^2x(A + Bx), where A and B are constants.
Solution:
Given eqn. of family of curves be
y = e^2x(A + Bx) … (1)
Diff. both sides w.r.t. x ; we have

which is the required differential equation.

Question 5.
Form the differential equation corres-ponding to y² – 2ay + x² = a² by eliminating a.
Solution:
Given differential eqn. be,
y² – 2ay + x² = a² …(1)
Diff. eqn. (1) both sides w.r.t. x ; we get

which is the required differential equation.

Question 6.
Find the differential equation of family of circles touching y-axis at the origin.
Solution:
The family of circles having radius a and touching y-axis at origin is given by (x – a)² + y² = a²
⇒ x² – 2ax + y² = 0 …(1)
where a be the arbitrary constant.
Diff. both sides eqn. (1) w.r.t. x ; we have
2x – 2a + 2y \(\frac { dy }{ dx }\) = 0
⇒ a = x + y \(\frac { dy }{ dx }\)
∴ from eqn. (1); we have
x² + y² – 2x(x + y\(\frac { dy }{ dx }\)) = o
⇒ y² – x² – 2xy\(\frac { dy }{ dx }\) = 0
⇒ 2xy\(\frac { dy }{ dx }\) + x² – y² = 0
which is the required diff. eqn.

Question 7.
Form the differential equation of the family of curves y = a sin (bx + c), a and c being arbitrary constants.
Solution:
Given eqn. of family of curves be
y = a sin (bx + c) …(1)
where a and c are arbitrary constants.
Diff. eqn. (1) both sides w.r.t. x ; we have
\(\frac { dy }{ dx }\) = a cos (bx + c). b … (2)
diff. eqn. (2) both sides w.r.t. x ; we have
\(\frac{d^2 y}{d x^2}\) = – a sin (bx + c)b²
= – b²y [using (1)]
⇒ \(\frac{d^2 y}{d x^2}\) + b²y = 0,
which is the required diff. eqn.

Question 8.
Find the differential equation repre-senting the family of curves given by y = Ax + \(\frac { B }{ x }\) where A and B are constants.
Solution:
Given eqn. of family of curves be,
y = Ax + \(\frac { B }{ x }\) … (1)
where A and B arbitrary constants
Diff. eqn. (1) both sides w.r.t. x, we get
\(\frac { dy }{ dx }\) = A – \(\frac { B }{ x² }\) … (2)
Multiply eqn. (2) by x and adding to eqn. (1); we get
x\(\frac{d y}{d x}+y=2 \mathrm{~A} x \Rightarrow \frac{d y}{d x}+\frac{y}{x}\) = 2A …(3)
Diff. eqn. (3) both sides w.r.t. x ; we get
\(\frac{d^2 y}{d x^2}+\frac{x \frac{d y}{d x}-y}{x^2}\) = 0
⇒ \(\frac{d^2 y}{d x^2}+\frac{1}{x} \frac{d y}{d x}-\frac{y}{x^2}\) = 0
be the required diff. eqn.

Question 9.
Form the differential equation of the family of curves represented by
c(y + c)² = x³.
Solution:
Given eqn. of family of curves be given by
c(y + c)² = x³ … (1)
Diff. eqn. (1) both sides w.r.t. x ; we get

which is the required diff. eqn.

Question 10.
Find the differential equation of the family of concentric circles x² + y² = a².
Solution:
Given eqn. of family of concentric circle be x² + y² = a² …(1)
Diff. (1) both sides w.r.t. x ; we have
2x + 2y\(\frac { dy }{ dx }\) = 0
⇒ x + y \(\frac { dy }{ dx }\) =0
which is the required differential equation.

Question 11.
Form the differential equation representing the family of curves, y =A cos 2x + B sin 2x, where A and B are constants.
Solution:
Given eqn. of family of curves be
y = A cos 2x + B sin 2x …(1)
Diff. both sides of eqn. (1) w.r.t. x; we have dy
\(\frac { dy }{ dx }\) = – 2A sin 2x + 2B cos 2x …(2)
Diff. eqn. (2) both sides w.r.t. x ; we have
\(\frac{d^2 y}{d x^2}\) = – 4A cos 2x – 4B sin 2x = – 4 [A cos 2x + B sin 2x] = – 4y [using eqn. (1)]
⇒ \(\frac{d^2 y}{d x^2}\) = 0,
which is the required diff. eqn.

Question 12.
Obtain the differential equation by eliminating ‘a’ and ‘b’ from equation y = e^x (a cos x + b sin x).
Solution:
Given eqn. be,
y = e^x (a cosx+ b sinx) …(1)
Diff. both sides eqn. (1) w.r.t. x ; we get

which is the required diff. eqn.

Question 13.
Form the differential equation of the family of curves represented by the equation (x – a)² + 2y² = a², where a is an arbitrary constant.
Solution:
Given eqn. of family of curve be

which is the required diff. eqn.

Question 14.
Form the differential equation representing family of ellipses having foci on X-axis and centre at the origin.
Solution:
The eqn. of family of ellipses having foci on x-axis and centre at origin be given by

which is the required differential equation.

Question 15.
Form a differential equation of the family of curves y² = 4ax.
Solution:
eqn. of family of curves be, y² = 4ax …(1)
diff. eqn. (1) both sides w.r.t. x; we have
2y\(\frac { dy }{ dx }\) = 4a … (2)
From (1) and (2); we have
\(y^2=2 x y \frac{d y}{d x} \quad \Rightarrow \frac{d y}{d x}=\frac{y}{2 x}\)
which is the required differential equation.