Category: Differential Equations, Integral Calculus, Analytic Geometry, Algebra
"Published in Newark, California, USA"
Find the equation of a curve having the given slope that passes through the indicated point:
Solution:
The slope of a curve is equal to the first derivative of a curve with respect to x. In this case, y' = dy/dx. Let's consider the given slope of a curve
Multiply both sides of the equation by dx, we have
Integrate on both sides of the equation, we have
In order to get the value of arbitrary constant, substitute the value of the given point which is P(5, 4) to the above equation, we have
Therefore, the equation of a curve is
Category: Differential Equations, Integral Calculus, Algebra
"Published in Newark, California, USA"
Find the general solution for
Solution:
Consider the given equation above
The
given equation is a 2nd Order Differential Equation because the second
derivative of y with respect to x is involved. We can rewrite given
equation as follows
Multiply both sides of the equation by dx, we have
Integrate on both sides of the equation, we have
Since the degree of a variable for both numerator and denominator are the same, then we have to do the division of a polynomial with another polynomial. After the division, the right side of the equation becomes
Multiply both sides of the equation by dx, we have
Integrate on both sides of the equation, we have
Consider
If
then
If
then
Hence, by integration by parts
Since
the degree of a variable for both numerator and denominator are the
same, then we have to do the division of a polynomial with another
polynomial. After the division, the right side of the equation becomes
Substitute the above equation to the original equation, we have
