CALCULUS

Integration by partial fractions is an integration technique which uses partial fraction decomposition to simplify the integrand. The integrand is written as partial fractions and then evaluated using standard methods. The integrals of many rational functions lead to a natural log function with absolute value expressions. This video explains what to do when you have non repeated linear factors factors.

To solve any Second Order Linear Homogeneous Differential Equation, first this you need to do, is to transform the equation in to an auxiliary or characteristics equation in the form: ar²+be+c=0 We have already seen how to do that in our previous lesson. The next move is to solve for r which are the roots of the equation (r intercept). Then determine the nature of the roots and substitute in to the following equations, depending on the nature of roots. y=C₁eʳ¹ˣ+C₂eʳ²ˣ. when you obtain real and distinct roots y=(C₁+C₂x)eʳˣ when you Obtain real and equal roots. and finally, if you Obtain a complex solution in the form: r= m+si or r = m-si where i is imaginary number, and m and s are real numbers, then y=eᵐˣ[C₁cos(st)+C₂sin(st)]

To solve any Second Order Linear Homogeneous Differential Equation, first this you need to do, is to transform the equation in to an auxiliary or characteristics equation in the form: ar²+be+c=0 We have already seen how to do that in our previous lesson. The next move is to solve for r which are the roots of the equation (r intercept). Then determine the nature of the roots and substitute in to the following equations, depending on the nature of roots. y=C₁eʳ¹ˣ+C₂eʳ²ˣ. when you obtain real and distinct roots y=(C₁+C₂x)eʳˣ when you Obtain real and equal roots. and finally, if you Obtain a complex solution in the form: r= m+si or r = m-si where i is imaginary number, and m and s are real numbers, then y=eᵐˣ[C₁cos(st)+C₂sin(st)]