Regular engagement with ISC Class 12 OP Malhotra Solutions Chapter 17 Differential Equations Ex 17(c) can boost students confidence in the subject.

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

Solve the following differential equations:

Question 1.
\(\frac { dy }{ dx }\) = 5x + 7
Solution:
OP Malhotra Class 12 Maths Solutions Chapter 17 Differential Equations Ex 17(c) 1

Question 2.
\(\frac { dy }{ dx }\) = sin x – x
Solution:
Given \(\frac { dy }{ dx }\) = sin x – x
⇒ dy = (sin x – x)dx;
On integrating ; we have
∫ dy = ∫ (sin x – x) dx
⇒ y = – cos x – \(\frac { x² }{ 2 }\) + C,
which is the required solution.

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

Question 3.
\(\frac { dy }{ dx }\) = x log x
Solution:
OP Malhotra Class 12 Maths Solutions Chapter 17 Differential Equations Ex 17(c) 2

Question 4.
\(\frac { dy }{ dx }\) + 2x = e3x
Solution:
Given \(\frac { dy }{ dx }\) + 2x = e3x
⇒ dy = (e3x – 2x) dx ;
On integrating ; we have
∫ dy = ∫ (e3x – 2x) dx
⇒ y = \(\frac{e^{3 x}}{3}\) – x² + C,
which is the required solution.

Question 5.
(x + 1)\(\frac { dy }{ dx }\) = x²
Solution:
OP Malhotra Class 12 Maths Solutions Chapter 17 Differential Equations Ex 17(c) 3

Question 6.
(x + 1)²\(\frac { dy }{ dx }\) = xex
Solution:
OP Malhotra Class 12 Maths Solutions Chapter 17 Differential Equations Ex 17(c) 4

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

Question 7.
\(\frac { dy }{ dx }\) = sin³ x cos² x + xex
Solution:
OP Malhotra Class 12 Maths Solutions Chapter 17 Differential Equations Ex 17(c) 5

Question 8.
\(\frac{d y}{d x}=\frac{1}{\sin ^4 x+\cos ^4 x}\)
Solution:
OP Malhotra Class 12 Maths Solutions Chapter 17 Differential Equations Ex 17(c) 6

Question 9.
\(\frac{d y}{d x}=x \sin ^2 x+\frac{1}{x \log x}\)
Solution:
OP Malhotra Class 12 Maths Solutions Chapter 17 Differential Equations Ex 17(c) 7

Question 10.
\(\sqrt{a+x}\)dy + xdx = 0
Solution:
Given diff. eqn. be,
OP Malhotra Class 12 Maths Solutions Chapter 17 Differential Equations Ex 17(c) 8
which is the required solution.

Question 11.
\(\frac{d y}{d x}=\sqrt{4-y^2}\)
Solution:
OP Malhotra Class 12 Maths Solutions Chapter 17 Differential Equations Ex 17(c) 9

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

Question 12.
\(\frac { dy }{ dx }\) = sec y
Solution:
Given \(\frac { dy }{ dx }\) = sec y ⇒ \(\frac{d y}{d x}=\frac{1}{\cos y}\)
⇒ cos y dy = dx
On integrating ; we have
∫ cos y dy = ∫ dx
⇒ sin y = x + c
be the required solution.

Question 13.
\(\frac{d y}{d x}=2^{-y}\)
Solution:
Given \(\frac{d y}{d x}=2^{-y}\)
⇒ \(\frac{1}{2^{-y}}\) dy = dx
⇒ 2y dy = dx
On integrating, we have
\(\int 2^y d y=\int d x \Rightarrow \frac{2^y}{\log 2}=x+\frac{c}{\log 2}\)
⇒ 2y = x log2 + c
which is the required solution.

Question 14.
Find the particular solution of edy/dx = x + 1, given that x = 0, y = 3.
Solution:
Give diff. eqn. be \(e \frac{d y}{d x}\) = x + 1
Taking logarithm on both sides, we have
\(\frac { dy }{ dx }\) = log (x + 1) ⇒ dy = log (x + 1) dx
On integrating ; we have
OP Malhotra Class 12 Maths Solutions Chapter 17 Differential Equations Ex 17(c) 10
given x = 0, y = 3 ∴ from (1) ; we have
3 = log 1 – 0 + c ⇒ c = 3
Thus eqn. (1) becomes :
y = (x+ 1) log (x + 1) – x + 3
be the required solution.

Question 15.
Find the particular solution of the differential equation log(\(\frac { dy }{ dx }\)) = 3x + 4y, given that y = 0 when x = 0.
Solution:
Given diff. eqn. be,
OP Malhotra Class 12 Maths Solutions Chapter 17 Differential Equations Ex 17(c) 11
which is the required solution.

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