08_Ex 3.8

Exercise Questions

Questions	Question Links
Q1. Find that each of the following equations…	3.8 Q-1
Q2. $\frac{d y}{d x}=-y$	3.8 Q-2
Q3. $y d y+x d y=0$	3.8 Q-3
Q4. $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1-\mathrm{x}}{\mathrm{y}}$	3.8 Q-4
Q5. $\frac{d y}{d x}=\frac{y}{x^{2}}$ $(y>0)$	3.8 Q-5
Q6. $\operatorname{Sin} y . \operatorname{Cosec} \cdot x \frac{d y}{d x}=1$	3.8 Q-6
Q7. $x d y+y(x-1) d x=0$	3.8 Q-7
Q8. $\frac{x^{2}+1}{y+1}=\frac{x}{y} \frac{d y}{d x}$	3.8 Q-8
Q9. $\frac{1}{x} \frac{d y}{d x}=\frac{1}{2}\left(1+y^{2}\right)$	3.8 Q-9
Q10. $2 \mathrm{x}^{2} \mathrm{y} \frac{d y}{\mathrm{dx}}=\mathrm{x}^{2}-1$	3.8 Q-10
Q11. $\frac{d y}{d x}+\frac{2 x y}{2 y+1}=x$	3.8 Q-11
Q12. $\left(x^{2}-x^{2} y\right) \frac{d y}{d x}+y^{2}+x y^{2}=0$	3.8 Q-12
Q13. $\operatorname{Sec}^{2} \mathrm{x} \tan \mathrm{ydx}+\operatorname{Sec}^{2} \mathrm{y} \tan \mathrm{x} d y=0$	3.8 Q-13
Q14. $y-x \frac{d y}{d x}=2\left(y^{2}+\frac{d y}{d x}\right)$	3.8 Q-14
Q15. $1+\operatorname{Cos} x \tan y \frac{d y}{d x}=0$	3.8 Q-15
Q16. $y-x \frac{d y}{d x}=3\left(1+x \frac{d y}{d x}\right)$	3.8 Q-16
Q17. $\operatorname{Sec} x+\tan y \frac{d y}{d x}=0$	3.8 Q-17
Q18. $\left(e^{x}+e^{-x}\right) \frac{d y}{d x}=e^{x}-e^{-x}$	3.8 Q-18
Q19. Find the solution of the equation $\frac{d y}{d x}-x=x y^{2}$.	3.8 Q-19
Q20. Solve the differential of $\frac{d y}{d t}=2 x$	3.8 Q-20
Q21. Solve the differential equation $\frac{\mathrm{dS}}{\mathrm{dt}}+2 S t=0$.	3.8 Q-21
Q22. In a culture, bacteria increase…	3.8 Q-22
Q23. A ball is thrown vertically upward with…	3.8 Q-23

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Overview

In this exercise, we explore several differential equations and their corresponding solutions. Each equation presents a unique form, allowing us to apply techniques like separation of variables, differentiation, and integration to verify solutions and understand the properties of the functions involved.

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Key Concepts

• Differential Equations: An equation that involves derivatives of a function and describes how the function changes with respect to one or more variables.
• Separation of Variables: A method used to solve differential equations by rearranging terms to isolate each variable on different sides of the equation.
• Verification of Solutions: Checking if a given function satisfies the original differential equation.

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Important Formulas

1. Separation of Variables:
   $\frac{dy}{dx} = \frac{g(x)}{h(y)} \quad \Rightarrow \quad h(y) , dy = g(x) , dx$
2. General Solution for Exponential Growth/Decay:
   $y = C e^{kx}$
3. Trigonometric Integration:
   $\int \sin(x) , dx = -\cos(x) + C$

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Tips and Tricks

• Simplification: Always try to simplify the differential equation by factoring or making substitutions before attempting integration.
• Check Units: Ensure the units on both sides of the equation match when verifying solutions.
• Use of Constants: When solving, remember to add the constant of integration after performing indefinite integration.

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Summary

This exercise demonstrates methods for solving first-order differential equations, including separation of variables and direct differentiation. The key is to recognize the structure of the equation, separate the variables where possible, and integrate both sides to find the general solution. Solutions are verified by substitution back into the original equation.

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Reference

By Sir Shahzad Sair:

By Great Science Academy:

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