Differential Equations Class 12 Maths 2 Exercise 6.4 Solutions Maharashtra Board

Balbharti 12th Maharashtra State Board Maths Solutions Book Pdf Chapter 6 Differential Equations Ex 6.4 Questions and Answers.

12th Maths Part 2 Differential Equations Exercise 6.4 Questions And Answers Maharashtra Board

I. Solve the following differential equations:

Question 1.
\(x \sin \left(\frac{y}{x}\right) d y=\left[y \sin \left(\frac{y}{x}\right)-x\right] d x\)
Solution:
Maharashtra-Board-12th-Maths-Solutions-Chapter-6-Differential-Equations-Ex-6.4-Q1

Question 2.
(x 2 + y 2 ) dx – 2xy . dy = 0
Solution:
(x 2 + y 2 ) dx – 2xy dy = 0
∴ 2xy dy = (x 2 + y 2 ) dx
∴ \(\frac{d y}{d x}=\frac{x^{2}+y^{2}}{2 x y}\) ………(1)
Maharashtra-Board-12th-Maths-Solutions-Chapter-6-Differential-Equations-Ex-6.4-Q2
Maharashtra-Board-12th-Maths-Solutions-Chapter-6-Differential-Equations-Ex-6.4-Q2.1
Maharashtra-Board-12th-Maths-Solutions-Chapter-6-Differential-Equations-Ex-6.4-Q2.2

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Question 3.
\(\left(1+2 e^{\frac{x}{y}}\right)+2 e^{\frac{x}{y}}\left(1-\frac{x}{y}\right) \frac{d y}{d x}=0\)
Solution:
Maharashtra-Board-12th-Maths-Solutions-Chapter-6-Differential-Equations-Ex-6.4-Q3
Maharashtra-Board-12th-Maths-Solutions-Chapter-6-Differential-Equations-Ex-6.4-Q3.1

Question 4.
y 2 dx + (xy + x 2 ) dy = 0
Solution:
y 2 dx + (xy + x 2 ) dy = 0
∴ (xy + x 2 ) dy = -y 2 dx
∴ \(\frac{d y}{d x}=\frac{-y^{2}}{x y+x^{2}}\) ……..(1)
Put y = vx
∴ \(\frac{d y}{d x}=v+x \frac{d v}{d x}\)
Substituting these values in (1), we get
Maharashtra-Board-12th-Maths-Solutions-Chapter-6-Differential-Equations-Ex-6.4-Q4
Maharashtra-Board-12th-Maths-Solutions-Chapter-6-Differential-Equations-Ex-6.4-Q4.1

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Question 5.
(x 2 – y 2 ) dx + 2xy dy = 0
Solution:
Maharashtra-Board-12th-Maths-Solutions-Chapter-6-Differential-Equations-Ex-6.4-Q5
Maharashtra-Board-12th-Maths-Solutions-Chapter-6-Differential-Equations-Ex-6.4-Q5.1
Maharashtra-Board-12th-Maths-Solutions-Chapter-6-Differential-Equations-Ex-6.4-Q5.2

Question 6.
\(\frac{d y}{d x}+\frac{x-2 y}{2 x-y}=0\)
Solution:
Maharashtra-Board-12th-Maths-Solutions-Chapter-6-Differential-Equations-Ex-6.4-Q6
Maharashtra-Board-12th-Maths-Solutions-Chapter-6-Differential-Equations-Ex-6.4-Q6.1
Maharashtra-Board-12th-Maths-Solutions-Chapter-6-Differential-Equations-Ex-6.4-Q6.2

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Question 7.
\(x \frac{d y}{d x}-y+x \sin \left(\frac{y}{x}\right)=0\)
Solution:
Maharashtra-Board-12th-Maths-Solutions-Chapter-6-Differential-Equations-Ex-6.4-Q7
Maharashtra-Board-12th-Maths-Solutions-Chapter-6-Differential-Equations-Ex-6.4-Q7.1

Question 8.
\(\left(1+e^{\frac{x}{y}}\right) d x+e^{\frac{x}{y}}\left(1-\frac{X}{y}\right) d y=0\)
Solution:
Maharashtra-Board-12th-Maths-Solutions-Chapter-6-Differential-Equations-Ex-6.4-Q8
Maharashtra-Board-12th-Maths-Solutions-Chapter-6-Differential-Equations-Ex-6.4-Q8.1

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Question 9.
\(y^{2}-x^{2} \frac{d y}{d x}=x y \frac{d y}{d x}\)
Solution:
Maharashtra-Board-12th-Maths-Solutions-Chapter-6-Differential-Equations-Ex-6.4-Q9
Maharashtra-Board-12th-Maths-Solutions-Chapter-6-Differential-Equations-Ex-6.4-Q9.1

Question 10.
xy \(\frac{d y}{d x}\) = x 2 + 2y 2 , y(1) = 0
Solution:
Maharashtra-Board-12th-Maths-Solutions-Chapter-6-Differential-Equations-Ex-6.4-Q10
Maharashtra-Board-12th-Maths-Solutions-Chapter-6-Differential-Equations-Ex-6.4-Q10.1

Maharashtra-Board-Solutions

Question 11.
x dy + 2y · dx = 0, when x = 2, y = 1
Solution:
∴ x dy + 2y · dx = 0
∴ x dy = -2y dx
∴ \(\frac{1}{y} d y=\frac{-2}{x} d x\)
Integrating, we get
Maharashtra-Board-12th-Maths-Solutions-Chapter-6-Differential-Equations-Ex-6.4-Q11
This is the general solution.
When x = 2, y = 1, we get
4(1) = c
∴ c = 4
∴ the particular solution is x 2 y = 4.

Question 12.
x 2 \(\frac{d y}{d x}\) = x 2 + xy + y 2
Solution:
x 2 \(\frac{d y}{d x}\) = x 2 + xy + y 2
∴ \(\frac{d y}{d x}=\frac{x^{2}+x y+y^{2}}{x^{2}}\) ………(1)
Put y = vx
∴ \(\frac{d y}{d x}=v+x \frac{d v}{d x}\)
Maharashtra-Board-12th-Maths-Solutions-Chapter-6-Differential-Equations-Ex-6.4-Q12

Maharashtra-Board-Solutions

Question 13.
(9x + 5y) dy + (15x + 11y) dx = 0
Solution:
(9x + 5y) dy + (15x + 11y) dx = 0
∴ (9x + 5y) dy = -(15x + 11y) dx
∴ \(\frac{d y}{d x}=\frac{-(15 x+11 y)}{9 x+5 y}\) ………(1)
Put y = vx
∴ \(\frac{d y}{d x}=v+x \frac{d v}{d x}\)
Maharashtra-Board-12th-Maths-Solutions-Chapter-6-Differential-Equations-Ex-6.4-Q13
Maharashtra-Board-12th-Maths-Solutions-Chapter-6-Differential-Equations-Ex-6.4-Q13.1
Maharashtra-Board-12th-Maths-Solutions-Chapter-6-Differential-Equations-Ex-6.4-Q13.2

Question 14.
(x 2 + 3xy + y 2 ) dx – x 2 dy = 0
Solution:
(x 2 + 3xy + y 2 ) dx – x 2 dy = 0
∴ x 2 dy = (x 2 + 3xy + y 2 ) dx
∴ \(\frac{d y}{d x}=\frac{x^{2}+3 x y+y^{2}}{x^{2}}\) ………(1)
Maharashtra-Board-12th-Maths-Solutions-Chapter-6-Differential-Equations-Ex-6.4-Q14
Maharashtra-Board-12th-Maths-Solutions-Chapter-6-Differential-Equations-Ex-6.4-Q14.1

Maharashtra-Board-Solutions

Question 15.
(x 2 + y 2 ) dx – 2xy dy = 0.
Solution:
Maharashtra-Board-12th-Maths-Solutions-Chapter-6-Differential-Equations-Ex-6.4-Q15
Maharashtra-Board-12th-Maths-Solutions-Chapter-6-Differential-Equations-Ex-6.4-Q15.1
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