IZTECH Math255 Notes p2

By E. Z.

Non Linear DE's

Bernouilli DE

Definition
If a DE is in the form of: \(\ \small y' + p(x)y=q(x)y^n; n ≠ 0, 1 \) it is called Bernouilli DE.

Solution Strategy of Bernouilli DE

\(\ y'+p(x)y=q(x)y^n \)

\(\ y'y^{-n}+p(x)y^{1-n}=q(x) \)

\(\ z=y^{1-n}, z'=(1-n)y^{-n}y' \)

\(\ z'+p(x)(1-n)z=q(x)(1-n) \)

Riccati DE

Definition
If a DE is in the form of: \(\ \small y' = r(x) + p(x) y + q(x) y^2 \) is called Riccati DE.

Solution Strategy of Riccati DE

For a known particular solution \(\ y_1 \)

\(\ y' = r(x) + p(x) y + q(x) y^2 \)

\(\ y(x)=y_1(x)+ \frac{1}{z(x)}, y'(x)=y_1'(x)-\frac {z'(x)}{z^2(x)} \)

\(\ z'+z[p(x)+2q(x)y_1]=-q(x) \)

and then transforming it back into 𝓎 from 𝓏

Exact DE

Definition
For DE in the form: \(\ M(x,y)dx+N(x,y)dy=0 \), Iff \(\ M_y(x,y)=N_x(x,y) \) called exact.

Solution Strategy for Exact DE

\(\ \frac{∂ φ(x,y)}{∂x} = M(x,y) \)

\(\ ∫M(x,y)dx+g(y)=ψ(x,y) \)

\(\ \frac{∂ ψ}{∂ y}=N(x,y) \)

Existance & Uniqueness

Linear Equations

Let 𝓅 and 𝓆 are continous functions on 𝕀: α < 𝓍 < β, \(\ 𝓍_0 \) ∈ (α , β) . Then there exist a UNIQUE solution \(\ \small φ ( 𝓍 ) \) such satisfies \(\ \small 𝓎' + 𝓅(𝓍)𝓎=𝓆(𝓍) \). For every 𝓍 ∈ 𝕀 and also satisfies 𝓎( \(\ 𝓍_0 \) ) = \(\ 𝓎_0 \).

Non-Linear Equations

Let 𝒻 &\(\ \small \frac{∂ 𝒻}{∂ 𝓎} \) be continous in α < 𝓍 < β & γ < 𝓎 < δ containing the point \(\ (𝓍_0 , 𝓎_0) \), Then on that region there exists a unique solution to the IVP:
\(\ 𝓎'=𝒻(𝓍, 𝓎) \)
\(\ 𝓎(𝓍_0)=𝓎_0 \)

Second Order Linear Eq.

General form of 2nd order ODE is: \(\ F(x,y,y',y'') \)


General form of 2nd order LINEAR ODE is: \(\ y''+p(x)y'+q(x)y=g(x) \)

So that linear equations are easy to solve unlike non-linear equations we can approximate non-linear terms to linear terms for some values of variable.

\(\ y''+sin(y)=0 \) is not linear
\(\ y''+y=0 \) is linear and an approximation for the equation above for x values close to 0

Linear Homogenous Eq.

Existance & Uniqueness

For first order linear homogenous equation: \(\ y'+p(x)y = 0 \); If function \(\ φ \) is a solution, then \(\ 𝒸.φ \) is a solution too. Where 𝒸 ∈ ℝ.
For second order linear homogenous equation: \(\ y''+p(x)y'+q(x)y = 0 \);

There is needed 2 concepts to determine the general solution:
-Superposition principle
-Linear dependence of function

Superposition

\(\ y''-y=0 \)

\(\ y_1=c_1 e^x \) \(\ y_2=c_2 e^{-x} \)
\(\ y_g=c_1 e^x + c_2 e^{-x} \)
\(\ c_1,c_2 ∈ ℝ \)

Linear Dependence of Two Functions

\(\ y_1 \) & \(\ y_2 \) are defined in \(\ 𝕀 ⊂ ℝ \) are said to be linearly dependent if:
\(\ y_1=𝒸y_2 \) where \(\ ∀ x ∈ 𝕀 , 𝒸 ∈ ℝ\).

\(\ sin(x) \), \(\ 2.sin(x) \) ; Linearly Dependent \(\ x \), \(\ x+3 \) ; Linearly Independent

Wronskian

Let 𝒻 & ℊ be continously differentiable eq.'s on \(\ 𝕀 ⊂ ℝ \) Then wronskian is:
𝒲(𝒻,ℊ) = \(\ \begin{vmatrix} f & f' \\ g & g' \end{vmatrix}\)

𝒲=0 ⇔ 𝒻 & ℊ are linearly dependent
𝒲≠ 0 ⇔ 𝒻 & ℊ are linearly independent
\(\ y''(x)+p(x)y'(x)+q(x)y(x)=0 \)

\(\ c_1y_1+c_2y_2=yg \) ⇔ ∃ \(\ x_0 ∈ 𝕀\), 𝒲(\(y_1, y_2) ≠ 0 \)

In order to find spesific \(\ c_1 \) & \(\ c_1 \) there is needed for 2 additional conditions.

Let 𝓅 & 𝓆 be continous functions on open interval \(\ 𝕀 \) containing \(\ x_0 \). Then there exist a unique solution to IVP:
\(\ y''+ 𝓅 (x)y'+ 𝓆 (x)y=0\)
\(\ y(x_0)=y_0 \), \(\ y'(x_0)=y_1 \)