Differential Equations (MTH401) VU
Lecture#1
Background
Linear y=mx+c
Quadratic ax2+bx+c=0
Cubic ax3+bx2+cx+d=0
Systems of Linear equations
ax+by+c=0
lx+my+n=0
Solution ?
Equation
Differential Operator
dy 1
=
dx x
Taking anti derivative on both sides
y=ln x
From the past
Algebra
Trigonometry
Calculus
Differentiation
Integration
Differentiation
• Algebraic Functions
• Trigonometric Functions
• Logarithmic Functions
• Exponential Functions
• Inverse Trigonometric Functions
More Differentiation
• Successive Differentiation
• Higher Order
• Leibnitz Theorem
Applications
• Maxima and Minima
1
© Copyright Virtual University of Pakistan
,Differential Equations (MTH401) VU
• Tangent and Normal
Partial Derivatives
y=f(x)
f(x,y)=0
z=f(x,y)
Integration
Reverse of Differentiation
By parts
By substitution
By Partial Fractions
Reduction Formula
Frequently required
Standard Differentiation formulae
Standard Integration Formulae
Differential Equations
Something New
Mostly old stuff
• Presented differently
• Analyzed differently
• Applied Differently
dy
− 5y =1
dx
( y − x ) dx + 4 xdy =0
3
d2y ⎛ dy ⎞
+ 5⎜ ⎟ − 4 y = ex
⎝ dx ⎠
2
dx
∂u ∂v
+ =0
∂y ∂x
∂u ∂v
x +y =u
∂x ∂y
∂ 2u ∂ 2u ∂u
− 2 +2 =0
∂x 2
∂t ∂t
2
© Copyright Virtual University of Pakistan
,Differential Equations (MTH401) VU
Lecture-2:
Fundamentals
Definition of a differential equation.
Classification of differential equations.
Solution of a differential equation.
Initial value problems associated to DE.
Existence and uniqueness of solutions
Elements of the Theory
Applicable to:
• Chemistry
• Physics
• Engineering
• Medicine
• Biology
• Anthropology
Differential Equation – involves an unknown function with one or more of its
derivatives
Ordinary D.E. – a function where the unknown is dependent upon only one
independent variable
Examples of DEs
dy
− 5y =1
dx
( y − x ) dx + 4 xdy =0
3
d2y ⎛ dy ⎞
+ 5⎜ ⎟ − 4 y = ex
⎝ dx ⎠
2
dx
∂u ∂v
+ =0
∂y ∂x
∂u ∂v
x +y =u
∂x ∂y
∂ 2u ∂ 2u ∂u
− 2 +2 =0
∂x 2
∂t ∂t
Specific Examples of ODE’s
3
© Copyright Virtual University of Pakistan
, Differential Equations (MTH401) VU
The order of an equation:
• The order of the highest derivative appearing in the equation
3
d2y ⎛ dy ⎞
+ 5⎜ ⎟ − 4 y = ex
⎝ dx ⎠
2
dx
∂ 4 y ∂ 2u
a 2
+ =0
∂x 4 ∂x 2
Ordinary Differential Equation
If an equation contains only ordinary derivatives of one or more dependent variables,
w.r.t a single variable, then it is said to be an Ordinary Differential Equation (ODE). For
example the differential equation
3
d2y ⎛ dy ⎞
+ 5 ⎜ ⎟ − 4 y = ex
⎝ dx ⎠
2
dx
is an ordinary differential equation.
Partial Differential Equation
4
© Copyright Virtual University of Pakistan
Lecture#1
Background
Linear y=mx+c
Quadratic ax2+bx+c=0
Cubic ax3+bx2+cx+d=0
Systems of Linear equations
ax+by+c=0
lx+my+n=0
Solution ?
Equation
Differential Operator
dy 1
=
dx x
Taking anti derivative on both sides
y=ln x
From the past
Algebra
Trigonometry
Calculus
Differentiation
Integration
Differentiation
• Algebraic Functions
• Trigonometric Functions
• Logarithmic Functions
• Exponential Functions
• Inverse Trigonometric Functions
More Differentiation
• Successive Differentiation
• Higher Order
• Leibnitz Theorem
Applications
• Maxima and Minima
1
© Copyright Virtual University of Pakistan
,Differential Equations (MTH401) VU
• Tangent and Normal
Partial Derivatives
y=f(x)
f(x,y)=0
z=f(x,y)
Integration
Reverse of Differentiation
By parts
By substitution
By Partial Fractions
Reduction Formula
Frequently required
Standard Differentiation formulae
Standard Integration Formulae
Differential Equations
Something New
Mostly old stuff
• Presented differently
• Analyzed differently
• Applied Differently
dy
− 5y =1
dx
( y − x ) dx + 4 xdy =0
3
d2y ⎛ dy ⎞
+ 5⎜ ⎟ − 4 y = ex
⎝ dx ⎠
2
dx
∂u ∂v
+ =0
∂y ∂x
∂u ∂v
x +y =u
∂x ∂y
∂ 2u ∂ 2u ∂u
− 2 +2 =0
∂x 2
∂t ∂t
2
© Copyright Virtual University of Pakistan
,Differential Equations (MTH401) VU
Lecture-2:
Fundamentals
Definition of a differential equation.
Classification of differential equations.
Solution of a differential equation.
Initial value problems associated to DE.
Existence and uniqueness of solutions
Elements of the Theory
Applicable to:
• Chemistry
• Physics
• Engineering
• Medicine
• Biology
• Anthropology
Differential Equation – involves an unknown function with one or more of its
derivatives
Ordinary D.E. – a function where the unknown is dependent upon only one
independent variable
Examples of DEs
dy
− 5y =1
dx
( y − x ) dx + 4 xdy =0
3
d2y ⎛ dy ⎞
+ 5⎜ ⎟ − 4 y = ex
⎝ dx ⎠
2
dx
∂u ∂v
+ =0
∂y ∂x
∂u ∂v
x +y =u
∂x ∂y
∂ 2u ∂ 2u ∂u
− 2 +2 =0
∂x 2
∂t ∂t
Specific Examples of ODE’s
3
© Copyright Virtual University of Pakistan
, Differential Equations (MTH401) VU
The order of an equation:
• The order of the highest derivative appearing in the equation
3
d2y ⎛ dy ⎞
+ 5⎜ ⎟ − 4 y = ex
⎝ dx ⎠
2
dx
∂ 4 y ∂ 2u
a 2
+ =0
∂x 4 ∂x 2
Ordinary Differential Equation
If an equation contains only ordinary derivatives of one or more dependent variables,
w.r.t a single variable, then it is said to be an Ordinary Differential Equation (ODE). For
example the differential equation
3
d2y ⎛ dy ⎞
+ 5 ⎜ ⎟ − 4 y = ex
⎝ dx ⎠
2
dx
is an ordinary differential equation.
Partial Differential Equation
4
© Copyright Virtual University of Pakistan
