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Mathematical problems for self assessment

Sample Problems
FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS

 

1. For the following differential equations, state their order and whether they are linear or nonlinear equations:

(i) image placeholder

(ii) image placeholder

(iii) image placeholder

(iv) image placeholder

[Ans. (i) order 4, nonlinear, (ii) order 3, linear, (iii) order 1, nonlinear, (iv) order 1, linear].

 

2. Verify that the function image placeholder solves the differential equation

image placeholder

 

3. Determine the differential equations satisfied by the following functions:

(i) image placeholder

(ii) image placeholder

[Ans. (i) image placeholder (ii) image placeholder].

 

4. Find the solutions of the following differential equations by direct integration:

(i) image placeholder

(ii) image placeholder image placeholder,

(iii) image placeholder image placeholder

[Ans. (i) image placeholder, (ii) image placeholder , (iii) image placeholder].

 

5. Find, by separation of variables, the particular solutions of the differential equations:

(i) image placeholder, y(-2) =4,

(ii) image placeholder,image placeholder .

[Ans. (i) image placeholder, (ii) image placeholder].

 

6. Find, by separation of variables, the general solutions of the differential equations:

(i) image placeholder,

(ii) image placeholder.

[Ans. (i) image placeholder, (ii) image placeholder].

 

7. By finding an appropriate integrating factor determine the general solutions of the following linear ordinary differential equations:

(i) image placeholder,

(ii) image placeholder.

[Ans. (i) image placeholder, (ii) image placeholder].

 

8. By finding an appropriate integrating factor determine the particular solution of the linear ordinary differential equation

image placeholder,image placeholder .

[Ans. image placeholder].

 

9. In an image placeholdercircuit with applied voltage image placeholder,the instantaneous current i(t) in the circuit satisfies the first order ordinary differential equation

image placeholder

with initial condition image placeholder. Find the particular solution of this linear ODE.

[Ans. image placeholder].

 

10. (The Bernoulli Equation) Consider the nonlinear ordinary differential equation given by

image placeholder, for image placeholder

and appropriate functions image placeholder and image placeholder. This equation is called the Bernoulli equation and it has particular applications in fluid mechanics and civil engineering problems. Show that by making the substitution image placeholder, this equation can be expressed in the form

image placeholder,

which is a linear equation that can be solved in the standard way using the integrating factor technique.

Hence find the general solution of the Bernoulli equation

image placeholder.

[Ans.image placeholder ].