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General and Particular Solution
General Solution - The general solution to a differential equation is the most general form that the solution can take and doesn't take any initial conditions into account. It contains a number of independent variables equal to the order of the differential equation.
Particular Solution - The particular solution to a differential equation is the specific solution that satisfies both the differential equation and its given initial condition(s). It can be contained by solving the specific value of any arbitrary constant present.
SEPARATION OF VARIABLES
SEPARATION OF VARIABLES
An equation of the form M dx + N dy = 0 can be put in simpler form A(x) dx + B(y) dy = 0. Then, the general solution of differential equation can be obtained by means of integration provided below.
where c is any arbitrary constant.
Sample Problems
Answer/s:
CamScanner 05-26-2021 07.52.pdfHOMOGENEOUS FUNCTIONS
HOMOGENEOUS FUNCTIONS
Given that the coefficients M and N in an equation of order one,
M (x, y) dx + N (x, y) dy = 0
are both homogeneous functions and are of the same degree in x and y.
Steps for solving Homogeneous DE:
Use substitution by letting y = vx ( dy = vdx + xdv) or x = vy (dx = vdy + ydv)
The resulting differential equation contains v and x or v and y and is a variable separable differential equation
Integrate and substitute back to the original variables after evaluation
Sample Problems:
Answer/s:
CamScanner 05-26-2021 08.17.pdfEXACT DIFFERENTIAL EQUATIONS
EXACT DIFFERENTIAL EQUATIONS
Differential equations in the form M (x,y) dx + N (x, y) dy = 0 are said to be exact if it satisfies the exactness test
The following function formulas are used when solving exact DEs:
where F is the solution to exact DE.
Sample Problems:
Test each of the following equations for exactness and solve the equation.
Answer/s:
CamScanner 05-27-2021 20.04.pdfLINEAR DIFFERENTIAL EQUATIONS
LINEAR DIFFERENTIAL EQUATIONS
The general linear differential equation of order n is an equation that can be written in the form:
The equation in the form y' + P(x)y = Q(x) where P and Q are functions of x only is called linear differential equation since y and its derivatives are of the first degree. The solution for y' + P(x)y = Q(x) is obtained by multiplying throughout an integrating factor to become
Sample Problems:
Obtain the general solution of the following differential equations:
Answer/s:
CamScanner 05-27-2021 20.05.pdfBERNOULLI'S DIFFERENTIAL EQUATIONS
BERNOULLI'S DIFFERENTIAL EQUATIONS
Differential equations in the form y' + P(x)y = Q(x)y^n, where P(x) and Q(x) are continuous functions on the interval and n is a real number. Differential equations in these form are called Bernoulli Equations.The equation may be further simplified to a linear equation in standard form.
Therefore, the equation in z and x will be:
Hence, any Bernoulli equation can be solved with the aid of change of variable (except when n = 1, where the equation will be solved using variable separable).
Sample Problems:
Answer/s:
CamScanner 05-28-2021 09.50.pdfOTHER SUBSTITUTION METHODS
OTHER SUBSTITUTION METHODS
- INTEGRATING FOUND BY INSPECTION
CamScanner 05-29-2021 13.34.pdf2. DETERMINATION OF INTEGRATING FACTORS
2. DETERMINATION OF INTEGRATING FACTORS
CamScanner 05-29-2021 13.46.pdf3. substitution suggested by the equation
3. substitution suggested by the equation
CamScanner 05-29-2021 13.52.pdf4. COEFFICIENT LINEAR IN TWO VARIABLES
4. COEFFICIENT LINEAR IN TWO VARIABLES
CamScanner 05-31-2021 08.23.pdfreflection/s
reflection/s
Upon solving sets of first order differential equations, I have realized that there are no general formulas for first order but instead we should look for different methods on how to solve first-order DEs by looking at several cases discussed. These include the variable separable which could be solved simply by integration, homogeneous equation by substituting variables of y or x having the same degree in x and y, linear differential equations wherein integrating factor is used, Bernoulli differential equations which simultaneously uses substitution and integrating factors to solve the equation, and a couple of substitutions that can be used to solve differential equations. There are some differential equations that are too cumbersome to solve, such as in Bernoulli and other substitution methods ( Integrating found via inspection, Determination of Integrating factors and Coefficients Linear in Two Variables). But through substitution, we can transform complex non-linear/linear first-order differential equations in simpler terms. I have noticed that differential equations under this type can be reduced into homogeneous, linear, or variable-separable type.
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