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Verifying Solutions: Introduction To

1. The document verifies whether y = 3x^4 is a solution to the differential equation 6y' + 2y + y'' = 0. When the derivatives are substituted, it is found that y = 3x^4 is NOT a solution. 2. It then verifies whether y = x^4 is a solution to the differential equation 8y'' + y' + 2y = 2x^4 + 4x^3 + 96x^2. When the derivatives are substituted, it is found that y = x^4 IS a solution.

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Sam Ashley Dela Cruz
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998 views2 pages

Verifying Solutions: Introduction To

1. The document verifies whether y = 3x^4 is a solution to the differential equation 6y' + 2y + y'' = 0. When the derivatives are substituted, it is found that y = 3x^4 is NOT a solution. 2. It then verifies whether y = x^4 is a solution to the differential equation 8y'' + y' + 2y = 2x^4 + 4x^3 + 96x^2. When the derivatives are substituted, it is found that y = x^4 IS a solution.

Uploaded by

Sam Ashley Dela Cruz
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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VERIFYING SOLUTIONS

1. Verify that 𝑦 = 3𝑥 4 is a solution to the equation


6𝑦 ′ + 2𝑦 + 𝑦 ′′ = 0.
SOLUTION:
Find the first two derivatives.
𝑦 ′ = 12𝑥 3
𝑦′′ = 36𝑥 2
Now substitute y, y', and y'' into the differential equation.
6𝑦 ′ + 2𝑦 + 𝑦 ′′ = 0.
6(12𝑥 3 ) + 2(3𝑥 4 ) + 36𝑥 2 = 0.
72𝑥 3 + 6𝑥 4 + 36𝑥 2 = 0
6𝑥 4 + 72𝑥 3 + 36𝑥 2 = 0
NOT A SOLUTION

2. Verify that 𝑦 = 𝑥 4 is a solution to the equation


8𝑦 ′′ + 𝑦 ′ + 2𝑦 = 2𝑥 4 + 4𝑥 3 + 96𝑥 2
SOLUTION:
Find the first two derivatives. INTRODUCTION TO
y’ = 4x3
y’’ = 12x2
DIFFERENTIAL
EQUATION
Now substitute y, y', and y'' into the differential equation.
8𝑦 ′′ + 𝑦 ′ + 2𝑦 = 2𝑥 4 + 4𝑥 3 + 96𝑥 2
8 12𝑥 2 + 4𝑥 3 + 2 𝑥 4 = 2𝑥 4 + 4𝑥 3 + 96𝑥 2
96𝑥 2 + 4𝑥 3 + 2𝑥 4 = 2𝑥 4 + 4𝑥 3 + 96𝑥 2
2𝑥 4 + 4𝑥 3 + 96𝑥 2 = 2𝑥 4 + 4𝑥 3 + 96𝑥 2
SOLUTION
WHAT IS
DIFFERENTIAL EQUATION
A differential equation is an equation 1. ORDER
which contains one or more terms and the The order of a differential equation is the order
of the highest-ordered derivative in the equation
derivatives of one variable (i.e.,
dependent variable) with respect to the 2. DEGREE
other variable (i.e., independent variable) The degree of a differential equation is the
largest power or exponent of the highest-
dy/dx = f(x) ordered derivative present in the equation
Here “x” is an independent variable and 3. TYPE
“y” is a dependent variable A differential equation may be ordinary or partial
EXAMPLE: as to the type of derivatives or differentials
dy/dx = 2x dy/dx = 4x appearing in the equation, that is, if it contains
dy/dx = 10x + 9 dy/dx = ex-y + x3e-y ordinary derivatives and partial derivatives
dy/dx = 16x - 7 dy/dx = 9x2
A differential equation contains derivatives GIVEN ORDER DEGREE TYPE
which are either partial derivatives or
ordinary derivatives. The derivative 𝝏𝒙𝟗 𝝏𝒙 1 9 PDE
represents a rate of change, and the + =𝟎
𝝏𝟗 𝒚 𝝏𝒚
differential equation describes a
relationship between the quantity that is (2x4)3+(x4)2=0 3 4 ODE
continuously varying with respect to the
change in another quantity.

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