Balbharati Maharashtra State Board Std 12 Commerce Statistics Part 1 Digest Pdf Chapter 8 Differential Equation and Applications Ex 8.5 Questions and Answers.

## Maharashtra State Board 12th Commerce Maths Solutions Chapter 8 Differential Equation and Applications Ex 8.5

Solve the following differential equations.

Question 1.

\(\frac{d y}{d x}+y=e^{-x}\)

Solution:

\(\frac{d y}{d x}+y=e^{-x}\) …….(1)

This is the linear differential equation of the form

This is the general solution.

Question 2.

\(\frac{d y}{d x}\) + y = 3

Solution:

\(\frac{d y}{d x}\) + y = 3

This is the linear differential equation of the form

This is the general solution.

Question 3.

x\(\frac{d y}{d x}\) + 2y = x^{2} . log x.

Solution:

x\(\frac{d y}{d x}\) + 2y = x^{2} . log x

∴ \(\frac{d y}{d x}+\left(\frac{2}{x}\right) \cdot y=x \cdot \log x\) …….(1)

This is the linear differential equation of the form

This is the general solution.

Question 4.

(x + y)\(\frac{d y}{d x}\) = 1

Solution:

(x + y) \(\frac{d y}{d x}\) = 1

∴ \(\frac{d x}{d y}\) = x + y

∴ \(\frac{d x}{d y}\) – x = y

∴ \(\frac{d x}{d y}\) + (-1) x = y ……(1)

This is the linear differential equation of the form

This is the general solution.

Question 5.

y dx + (x – y^{2}) dy = 0

Solution:

y dx + (x – y^{2}) dy = 0

∴ y dx = -(x – y^{2}) dy

∴ \(\frac{d x}{d y}=-\frac{\left(x-y^{2}\right)}{y}=-\frac{x}{y}+y\)

∴ \(\frac{d x}{d y}+\left(\frac{1}{y}\right) \cdot x=y\) ……(1)

This is the linear differential equation of the form

This is the general solution.

Question 6.

\(\frac{d y}{d x}\) + 2xy = x

Solution:

\(\frac{d y}{d x}\) + 2xy = x ………(1)

This is the linear differential equation of the form

This is the general solution.

Question 7.

(x + a) \(\frac{d y}{d x}\) = -y + a

Solution:

(x + a) \(\frac{d y}{d x}\) + y = a

∴ \(\frac{d y}{d x}+\left(\frac{1}{x+a}\right) y=\frac{a}{x+a}\) ……..(1)

This is the linear differential equation of the form

This is the general solution.

Question 8.

dy + (2y) dx = 8 dx

Solution:

dy + (2y) dx = 8 dx

∴ \(\frac{d y}{d x}\) + 2y = 8 …….(1)

This is the linear differential equation of the form

This is the general solution.