Category: Differential Equations, Integral Calculus, Algebra
"Published in Newark, California, USA"
Find the general solution for the given equation
Solution:
Consider the given equation
Check if the given equation is exact or not exact as follows
Let
Let
Since
The given equation is not an exact equation. In this case, we need to find the integrating factor and then multiply it to the both sides of the equation. Let's consider again the given equation
Arrange the above equation into its standard form as follows
Divide both sides of the equation by dx
Divide both sides of the equation by (x + 1)2
As you notice that the above equation is now already in standard form
where
and
The integrating factor is
Therefore, the general solution for the above equation is

This website will show the principles of solving Math problems in Arithmetic, Algebra, Plane Geometry, Solid Geometry, Analytic Geometry, Trigonometry, Differential Calculus, Integral Calculus, Statistics, Differential Equations, Physics, Mechanics, Strength of Materials, and Chemical Engineering Math that we are using anywhere in everyday life. This website is also about the derivation of common formulas and equations. (Founded on September 28, 2012 in Newark, California, USA)
Saturday, January 19, 2013
Friday, January 18, 2013
Area Derivation - Triangle, Three Vertices
Category: Analytic Geometry, Plane Geometry, Algebra
"Published in Newark, California, USA"
Given a triangle with vertices as shown below
Prove that the area of a triangle is
Solution:
The first step is to draw vertical lines from the vertices to the x-axis as shown below
Area of Triangle ABC = Area of Trapezoid ACDF - Area of Trapezoid ABEF - Area of Trapezoid BCDE
Label further the figure, we have
Consider Trapezoid ACDF
Consider Trapezoid ABEF
Consider Trapezoid BCDE
Therefore
Using a distance of two points formula
The above equation becomes
You notice that the above equation looks like a matrix or determinant because of the sequence of x and y. The three positive terms are the principal diagonals (product from top left to bottom right) while the three negative terms are the secondary diagonals (product from bottom left to top right). The above equation can be written as
Therefore
"Published in Newark, California, USA"
Given a triangle with vertices as shown below
![]() |
Photo by Math Principles in Everyday Life |
Prove that the area of a triangle is
Solution:
The first step is to draw vertical lines from the vertices to the x-axis as shown below
![]() |
Photo by Math Principles in Everyday Life |
Label further the figure, we have
![]() |
Photo by Math Principles in Everyday Life |
Consider Trapezoid ACDF
Consider Trapezoid ABEF
Consider Trapezoid BCDE
Therefore
Using a distance of two points formula
The above equation becomes
You notice that the above equation looks like a matrix or determinant because of the sequence of x and y. The three positive terms are the principal diagonals (product from top left to bottom right) while the three negative terms are the secondary diagonals (product from bottom left to top right). The above equation can be written as
Therefore
Thursday, January 17, 2013
Proving - Parallelogram, Triangles
Category: Plane Geometry
"Published in Newark, California, USA"
Given: Parallelogram CDEF; S and T are midpoints of EF and ED.
Prove: SR ≅ FD
Solution:
Consider the given figure
Proof:
1. Statement: Parallelogram CDEF
Reason: Given item.
2. Statement: S and T are the midpoints of EF and ED.
Reason: Given item.
3. Statement: FS ≅ SE
Reason: Point S bisects the line segment EF into two equal parts.
4. Statement: ET ≅ TD
Reason: Point T bisects the line segment ED into two equal parts.
5. Statement: FE ║ CD
Reason: The opposite sides of a parallelogram are parallel.
6. Statement: ∠ SET ≅ ∠ TDR
Reason: If a transversal line (ED) passed the two parallel lines (EF and CD), then the alternating interior angles are congruent.
7. Statement: ∠ STE ≅ ∠ DTR
Reason: Vertical angles are congruent.
8. Statement: ∆SET ≅ ∆DTR
Reason: Angle Side Angle (ASA) Postulate.
9. Statement: DR ≅ SE
Reason: Since ∆SET ≅ ∆DTR, then all sides of a triangle are congruent to all sides of other triangle.
10. Statement: DR ≅ SE ≅ FS
Reason: Transitive property of congruence.
11. Statement: FS ║ DR
Reason: Since Point R is colinear with CD and CD is parallel to EF, then it follows that FS is parallel to DR since Point S is colinear with EF.
12. Statement: FDRS is a parallelogram.
Reason: Since FS and DR are parallel and congruent, then it follows that the figure formed by the points FDRS is a parallelogram.
13. Statement: FD ≅ SR
Reason: The two opposite sides of a parallelogram are congruent and parallel.
"Published in Newark, California, USA"
Given: Parallelogram CDEF; S and T are midpoints of EF and ED.
Prove: SR ≅ FD
![]() |
Photo by Math Principles in Everyday Life |
Solution:
Consider the given figure
![]() |
Photo by Math Principles in Everyday Life |
Proof:
1. Statement: Parallelogram CDEF
Reason: Given item.
2. Statement: S and T are the midpoints of EF and ED.
Reason: Given item.
3. Statement: FS ≅ SE
Reason: Point S bisects the line segment EF into two equal parts.
4. Statement: ET ≅ TD
Reason: Point T bisects the line segment ED into two equal parts.
5. Statement: FE ║ CD
Reason: The opposite sides of a parallelogram are parallel.
6. Statement: ∠ SET ≅ ∠ TDR
Reason: If a transversal line (ED) passed the two parallel lines (EF and CD), then the alternating interior angles are congruent.
7. Statement: ∠ STE ≅ ∠ DTR
Reason: Vertical angles are congruent.
8. Statement: ∆SET ≅ ∆DTR
Reason: Angle Side Angle (ASA) Postulate.
9. Statement: DR ≅ SE
Reason: Since ∆SET ≅ ∆DTR, then all sides of a triangle are congruent to all sides of other triangle.
10. Statement: DR ≅ SE ≅ FS
Reason: Transitive property of congruence.
11. Statement: FS ║ DR
Reason: Since Point R is colinear with CD and CD is parallel to EF, then it follows that FS is parallel to DR since Point S is colinear with EF.
12. Statement: FDRS is a parallelogram.
Reason: Since FS and DR are parallel and congruent, then it follows that the figure formed by the points FDRS is a parallelogram.
13. Statement: FD ≅ SR
Reason: The two opposite sides of a parallelogram are congruent and parallel.
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