TS Inter Second Year Maths 2B Differential Equations Important Questions Long Answer Type

Students must practice these Maths 2B Important Questions TS Inter Second Year Maths 2B Differential Equations Important Questions Long Answer Type to help strengthen their preparations for exams.

TS Inter Second Year Maths 2B Differential Equations Important Questions Long Answer Type

Model III – Problems on non-homogeneous D.E

Question 1.
Solve (2x + y + 1) dx + (4x + 2y – 1) dy = 0. [(TS) Mar. ’15]
Solution:

Question 2.
Solve the differential equation \(\frac{d y}{d x}=\frac{x+2 y+1}{2 x+4 y+3}\). [Mar. ’19 (TS)]
Solution:
Given differential equation is \(\frac{d y}{d x}=\frac{x+2 y+1}{2 x+4 y+3}\) ……(1)

4v + log (4v + 5) = 8x + 8c
4(x + 2y) – 8x + log [4(x + 2y) + 5] = c
4x + 8y – 8x + log(4x + 8y + 5) = c
8y – 4x + log (4x + 8y + 5) = c

Question 3.
Solve \(\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{x}-\mathrm{y}+3}{2 \mathrm{x}-2 \mathrm{y}+5}\). [(AP) May ’15]
Solution:

Question 4.
Solve \(\frac{d y}{d x}=\frac{4 x+6 y+5}{3 y+2 x+4}\). [(TS) May ’19]
Solution:



8z + 9 log|8z + 23| = 64x + c
8(2x + 3y) + 9 log|8(2x + 3y) + 23| = 64x + c
24y + 9 log|16x + 24y + 23| = 48x + c
Which is the required solution.

Question 5.
Solve \(\frac{d y}{d x}=\frac{3 y-7 x+7}{3 x-7 y-3}\)
Solution:

Question 6.
Solve \(\frac{d y}{d x}=\frac{x+2 y+3}{2 x+3 y+4}\)
Solution:

Question 7.
Solve the differential equation (x – y) dy = (x + y + 1) dx. [Mar. ’17 (TS)]
Solution:

Question 8.
Solve (2x + y + 3) dx = (2y + x + 1) dy
Solution:

Question 9.
Solve \(\frac{d y}{d x}=\frac{6 x+5 y-7}{2 x+18 y-14}\)
Solution:

Question 10.
Find the solution of the equation x(x – 2) \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = 2(x – 1)y = x³(x – 2), which satisfies the condition that y = 9, when x = 3. [(TS) May ’17]
Solution:

Question 11.
Form the differential equation corresponding to the family of circles of radius r given by (x – a)² + (y – b)² = r², where ‘a’ and ‘b’ are parameters.
Solution:
Given equation is (x – a)² + (y – b)² = r² ………(1)
a, b are parameters.
differentiating (1) w.r.t. ‘x’ on both sides we get
2(x – a)(1 – 0) + 2(y – b)(\(\frac{\mathrm{dy}}{\mathrm{dx}}\) – 0) = 0
(x – a) + (y – b)(\(\frac{\mathrm{dy}}{\mathrm{dx}}\)) = 0 ……..(2)
Again differentiating (2) w.r.t ‘x’ on both sides, we get



Which is the required differential equation.

Question 12.
Form the differential equation from y = c(x – c)², where ‘c’ is an arbitrary constant. [(TS) Mar. ’16]
Solution:
Given y = c(x – c)² ………(1)
(where ‘c’ is an arbitrary constant)
Diff. with respect to x on both sides