Partial Differential Equations Examples
Example 1: Given the function c = f(x2 – y2), find its partial differential equation.
Solution:
Differentiate both LHS and RHS w.r.t.x.
∂u/∂x = 2x.f'(x2 – y2)…(1)
∂u/∂y = -2y.f'(x2 – y2)…(2)
Dividing (1) by (2), we get
(∂u/∂x)/(∂u/∂y)= -x/y
Thus, differential equation is given as: y.∂u/∂x+ x.∂u/∂y = 0
Example 2: Prove that u(x,t) = sin(at)cos(x) is a solution to ????2u/????t2 = a2(????2u/????x2) , given that a is constant.
Solution:
Differentiate both LHS and RHS
∂u/∂t = acos(at)cos(x)
∂2u/∂t2 = -a2sin(at)cos(x)
Since,
- ux = – sin (at) sin (x)
- uxx = – sin (at)cos(x)
????2u/????t2 = a2(????2u/????x2)
Thus, u(x,t) = sin(at) cos(x) is a solution of ????2u/????t2 = a2(????2u/????x2)
Example 3: Form the partial differential equation for all such spheres having a center in the x-y plane and fixed radii.
Solution:
General equation of such spheres is, (x – a)2 + (y – b)2 + z2 = r2
Differentiate LHS and RHS w.r.t.x and w.r.t.y
2z{∂z/∂x} = -2(x – a)
2z{∂z/∂y} = -2(y – a)
(x – a) = -z{∂z/∂x}
(y – a) = -z{∂z/∂y}
Substituting these values in the general form of equation, the partial differential equation is,
[Tex]z^2 = \frac{r^{2}}{(\frac{\partial z}{\partial x})^{2} + (\frac{\partial z}{\partial y})^{2} + 1} [/Tex]
Example 4: Prove that ????2p/????t2 = b2????2p/????x2 if p(x, t) = sin(bt)cosx.
Solution:
????p/????t = b cos(bt) cos(x)
⇒ ????2p/????t2 = -b2 sin(bt) cos(x)
Now,
????p/????x = -sin(bt) sin(x)
⇒ ????2p/????x2 = -sin(bt) cos(x)
b2????2p/????x2 = -b2sin(bt) cos(x)
Hence proved.
Example 5: Reduce uxx + 5uxy + 6uyy = 0. to its canonical form and solve it.
Solution:
Since,
b2 − 4ac = 1 > 0 {for the given equation, it is hyperbolic}
Let,
- μ(x, y)=3x − y
- η(x, y)=2x − y
Then,
- μx = 3
- ηx = 2
- μy = −1
- ηy = −1
u = u(μ(x, y), η(x, y))
ux = uμμx + uηηx = 3uμ + 2uη
uy = uμμy + uηηy = −uμ − uη
uxx = (3uμ + 2uη)x = 3(uμμμx + uμηηx) + 2(uημμx + uηηηx)
uxx = 9uμμ + 12uμη + 4uηη…(1)
uxy = (3uμ + 2uη)y = 3(uμμμy + uμηηy) + 2(uημμy + uηηηy)
uxy = −3uμμ − 5uμη − 2uηη…(2)
uyy = −(uμ + uη)y = −(uμμμy + uμηηy + uημμy + uηηηy)
uyy = uμμ + 2uμη + uηη…(3)
Thus, canonical form is given as, uμη = 0
The general solution is, u(x, y) = F(3x − y) + G(2x − y)
Partial Differential Equation
Partial Differential Equation contains an unknown function of two or more variables and its partial derivatives with respect to these variables. The order of partial differential equations is that of the highest-order derivatives. Such equations aid in the relationship of a function with several variables to their partial derivatives. They are extremely important in analyzing natural phenomena such as sound, temperature, flow properties, and waves.
They are used to express issues that include an unknown function with numerous dependent and independent variables, as well as the second derivative of this function with respect to the independent variables.
In this article, we will learn the definition of Partial Differential Equations, their representation, their order, the types of partial differential equations, how to solve PDE, and many more details.
Table of Content
- What is a Partial Differential Equation?
- Partial Differential Equation Definition
- Order of Partial Differential Equation
- Examples of Partial Differential Equations
- Degree of Partial Differential Equation
- General Form of Partial Differential Equation
- Representing Partial Differential Equation
- Types of Partial Differential Equations
- First-Order Partial Differential Equation
- Second-Order Partial Differential Equation
- Quasi-Linear Partial Differential Equation
- Homogeneous Partial Differential Equation
- Classification of Partial Differential Equation
- Applications of Partial Differential Equations
- How to Solve Partial Differential Equations
- Steps for Solving Partial Differential Equations
- Partial Differential Equation Class 12
- Partial Differential Equations Examples
- Practice Questions on Partial Differential Equations
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