Differential Equations for Engineers

2nd Constant Homogeneous Differential Equation Solutions

Yp Case 1 f(x)= polynomial function

Yp Case 2 f(x)= e function

Yp Case 3 f(x)= sin(x) or cos(x)

Yp Case 4 f(x)=Addition and Subtraction Case 1, 2, 3

Yp Case 5 f(x)= e X polinom

Yp Case 6 f(x)= e X Sin(x) Cos(x)

Yp Case 7 f(x) polinom * (Sin(x) Cos(x))

Variation of parameter

y′′−2y′−15y=2e^4x

Variation of parameter

y′′−2y′−15y=2e^4x

Fast Ver.

High Order Differential Equation

High Order 4 Degrees Homogeneous Diff. Equ.

High Order Homogeneous Diff. Equ.

High Order 4 Degrees Homogeneous

High Order 5 Degrees Homogeneous Diff. Equ.

High Order Diff. Equ. Homogeneous 3nd Order

High Order Diff. Equ. Undetermined Coeff. Method

High Order Non Homogeneous Undetermined Coefficient Method

High Order Diff. Equ. Variation of parameter

Laplace transform

Definition of Laplace

Laplace Rules Constant Number

Laplace Rules t^n

Laplace Rules e^at

Laplace Rules Sin and Cos

Laplace Rules e^at*t and e^at*Sin(bt) or e^at*Cos(bt)

Laplace Rules t*Sin(bt) or t*Cos(bt)

Laplace Example Non Homogeneou 50t-150

Differential Systems

Eigen Values Eigen Vectors Fundamental

Eigen Values Eigen Vectors Opt-1

Eigen Values Eigen Vectors Opt-2

Eigen Values Eigen Vectors Opt-3

Non-Homogeneous Fundamental

Non-Homogeneous

PDE Partial Diferantial Equation

Example of PDE by using Separation of Variables

Differential Equations

Laplace e^(-t)Sin(t)

Laplace f(t) Step Function

Undetermined coefficient method Non Homegeneus Part (1+sinx)

Ordinary 2nd order Constant Coefficient Non homegeneus differential equations and solved by using Undetermined coefficient method. Non homegeneus part "1 + SinX"

We can see that homogeneus part solution roots x(1)(-1 +2i) and x(2)(-1-2i)

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Ordinary 2nd order Constant Coefficient Non homegeneus differential equations and solved by using Undetermined coefficient method. Non homegeneus part "xe^x-e^x"

We can see that homogeneus part solution roots x(1)=x(2)=1 but homogeneeus solution and non-homogeneus part of equation are similar. So we use yp=(Ax^3+Bx^2)e^x instead of using Axe^x

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Undetermined coefficient method Non Homegeneus (3Sin(2x))

Ordinary 2nd order Constant Coefficient Non homegeneus differential equations and solved by using Undetermined coefficient method. Non homegeneus part "3Sin(2x)"

We can see that homogeneus part solution roots x(1)=x(2)=2i but homogeneeus solution and non-homogeneus part of equation are similar. So we use yp=AxSin(2x)+BxCos(2x)

Thanks

Undetermined coefficient method Non Homegeneus (3Sin(2x)) Summary

Ordinary 2nd order Constant Coefficient Non homegeneus differential equations and solved by using Undetermined coefficient method. Non homegeneus part "3Sin(2x)"

We can see that homogeneus part solution roots x(1)=x(2)=2i but homogeneeus solution and non-homogeneus part of equation are similar. So we use yp=AxSin(2x)+BxCos(2x)

Thanks

High Order Diff Eq. Undetermined coefficient method

y^(3)-3y^''+2y^'=x+e^x

Constant coefficient 3nd differential equation

High Order Ordinary differential equation

Linear differential equation

Non-Homogeneous differential equation

Non-Constant coefficient 2nd Homogeneous ODE

x^2y''+3xy'+y=0

y1(x)=1/x

y=c1y1+c2y2

PDE Wave Non Homogeneous (Sin(piX)) Laplace transform

Understand the question

Use Laplace transform t to s domain and then get ODE

Solve ODE with ch. eq. for yh and  Undetermind coefficient method for yp and then turn to

s domail to t domain by using Reverse Laplace to get general solution

Differential Systems Non Homogeneous