In case an equation is in the form y = f(ax+by+c), i.e., the RHS is a linear function of x and y. We will use the substitution U = ax + by + c to find an implicit general solution. The right hand side of the following first order problem y = (4x - 3y + 1) 5/6 +, y(0) = 0 is a function of a linear combination of x and y, i.e., y = f(ax +by+c). To solve this problem we use the substitution v = ax + by + c which transforms the equation into a separable equation. We obtain the following separable equation in the variables x and v: U' = Solving this equation an implicit general solution in terms of x, v can be written in the form x+ Transforming back to the variables x and y the above equation becomes x+ Next using the initial condition y(0) = 0 we find C = 6 = C. v= = C. Then, after a little algebra, we can write the unique explicit solution of the initial value problem as
In case an equation is in the form y = f(ax+by+c), i.e., the RHS is a linear function of x and y. We will use the substitution U = ax + by + c to find an implicit general solution. The right hand side of the following first order problem y = (4x - 3y + 1) 5/6 +, y(0) = 0 is a function of a linear combination of x and y, i.e., y = f(ax +by+c). To solve this problem we use the substitution v = ax + by + c which transforms the equation into a separable equation. We obtain the following separable equation in the variables x and v: U' = Solving this equation an implicit general solution in terms of x, v can be written in the form x+ Transforming back to the variables x and y the above equation becomes x+ Next using the initial condition y(0) = 0 we find C = 6 = C. v= = C. Then, after a little algebra, we can write the unique explicit solution of the initial value problem as
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:In case an equation is in the form y = f(ax+by+c), i.e., the RHS is a linear function of x and y. We will use the substitution = ax + by + c to
find an implicit general solution.
The right hand side of the following first order problem
y = (4x − 3y + 1) 5/6 +, y(0) = 0
is a function of a linear combination of x and y, i.e., y = f(ax +by+c). To solve this problem we use the substitution v= ax + by + c which
transforms the equation into a separable equation.
We obtain the following separable equation in the variables x and v:
U' =
Solving this equation an implicit general solution in terms of x, u can be written in the form
x+
Transforming back to the variables x and y the above equation becomes
x+
= C.
y =
= C.
Next using the initial condition y(0) = 0 we find C = 6
Then, after a little algebra, we can write the unique explicit solution of the initial value problem as
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