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Ma102 Ode

This document provides an overview of topics covered in a tutorial on first order ordinary differential equations (ODEs). It includes instructions to: 1. Classify different types of differential equations and specify their order. 2. Consider the solution to the differential equation y' = θy, where θ is a constant. 3. Solve sample first order differential equations. The document covers various methods for solving first order ODEs, including reducing to separable variable form, using integrating factors, and Picard's method of successive approximations. It also includes how to find the differential equation corresponding to a given family of curves.

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94 views2 pages

Ma102 Ode

This document provides an overview of topics covered in a tutorial on first order ordinary differential equations (ODEs). It includes instructions to: 1. Classify different types of differential equations and specify their order. 2. Consider the solution to the differential equation y' = θy, where θ is a constant. 3. Solve sample first order differential equations. The document covers various methods for solving first order ODEs, including reducing to separable variable form, using integrating factors, and Picard's method of successive approximations. It also includes how to find the differential equation corresponding to a given family of curves.

Uploaded by

dundikuladeepeswar
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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MA102: First Order ODEs

Tutorial 1

1. Classify the following Differential equations: Linear/ Non-linear/ Ordinary/ Par-


tial, etc. and specify the order.

(a) y ′′ + 3y ′ + 20y = ex ,
p
(b) 1 + y 3 = x2 ,
(c) y ′′ + y 2 = cos(x),
(d) y ′ + xy = cos(y ′ ),
(e) (xy ′ )′ = xy,
(f) ux + uy = 0,
(g) uxx + uyy = ut .

2. Consider the differential equations y ′ = θy; x > 0; where θ is a constant. Show


that

(a) if ϕ(x) is any solution and ψ(x) = ϕ(x)e−θx , then ψ(x) is a constant;
(b) if θ < 0, then every solution tends to zero as x → ∞.

3. Solve the following Differential equations:

(a) x2 y ′ = 3(x2 + y 2 ) arctan xy + xy,




(b) y ′ = sin2 (x − y + 1).


ax+by+c
4. (a) Reduce the differential equation y ′ = f dx+ey+f

, ae − bd ̸= 0 to a separable
variable form. What happens if ae = bd ?
(b) Hence, find the general solution of the following differential equations:
i. (x + 2y + 1) − (2x + y − 1)y ′ = 0,
(8x−2y+1)2
ii. y ′ = (4x−y−1)2
.

5. By making a substitution v = xyn or y = vxn and choosing suitable n, show that


the following differential equations can be transformed into separable variables,
and hence solve them:
1−xy 2
(a) y ′ = 2x2 y
,
2+3xy 2
(b) y ′ = 4x2 y
.

6. Show that the following equations are exact and, hence, find their general solution:

(a) (cos(x) cos(y) − cot(x))dx − (sin(x) sin(y))dy = 0,


2 2
(b) 2x(y + 3x − ye−x )dx + (x2 + 3y 2 + e−x )dy = 0.
(c) (x + 2y − 3)dy − (2x − y + 1)dx = 0.
(d) y(sin 2x)dx − (1 + y 2 + cos2 x)dy = 0

7. Show that 2 sin(y 2 )dx + xy cos(y 2 )dy = 0 admits an integrating factor which is a
function of x only. Also, solve the differential equation.
8. A. Consider homogeneous equation M (x, y)dx + N (x, y)dy = 0. If Mx + Ny ̸= 0,
1
then Mx +N y
is an integrating factor of the differential equation. Using this,
solve the following equation:

(4y 2 + 3xy)dx − (3xy + 2x2 )dy = 0.

B. Further, multiply with xα y β and find α and β such that the corresponding
equation becomes exact. Hence, find the solution. Compare both solutions!

9. Reduce the following differential equations into linear form and solve:
y3
(a) y 2 y ′ + x
= sin(x),
(b) y ′ sin(y) + x cos(y) = x,
(c) y ′ = y(xy 3 − 1),
(d) (ey − 2xy)y ′ = y 2 ,
(e) y − xy ′ = y ′ y 2 ey .

10. Use Picard’s method of successive approximation to solve the following initial value
problems and compare these results with the exact solutions:
√
(a) y ′ = 2 x; y(0) = 1,
(b) y ′ + xy = x; y(0) = 0,
(c) y ′ = 2 y3 ; y(0) = 0.
p

11. Find the differential equation corresponding to the following family of curves:

(a) xy 2 − 1 = cy,
(b) y = ax2 + be2x ,
(c) y = a sin(x) + b cos(x) + b.

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