In: Advanced Math

Consider the initial value problem: y0 = 3 + x−y, y(0) = 1 (a)
Solve it analytically. (b) Solve it using Euler’s method using step
size h = 0.1 and ﬁnd an approximation to true solution at x = 0.3.
(c) What is the error in the Euler’s method at x = 0.3

Solve the initial value problem:
y'' + y = cos(x)
y(0) = 2
y'(0) = -3
y' being the first derivative of y(x), y'' being the second
derivative, etc.

Solve the following initial value problem using Laplace
transforms:
y000 + y0 = et, y(0) = y0(0) = y00(0) = 0.

Solve the initial value problem: y''+2y'+y = x^2 , y(0)=0 ,
y'(0) = 0

Solve the initial value problem. y'=(y^2)+(2xy)+(x^2)-(1),
y(0)=1

Consider the initial value problem
y′ = 18x − 3y, y(0) = 2
(a) Solve it as a linear 1st order ODE with the method of the
integrating factor.
(b) Solve it using a substitution method.
(c) Solve it using the Laplace transform.

Solve the following initial value problem.
y(4) − 5y′′′ + 4y′′ =
x, y(0) = 0,
y′(0) = 0,
y′′(0) = 0,
y′′′(0) = 0.

Use power series to solve the initial value problem
x^2y''+xy'+x^2y=0, y(0)=1, y'(0)=0

Solve the given initial-value problem.
y'' + y = 0, y(π/3) =
0, y'(π/3) = 4

(3 pts) Solve the initial value problem
25y′′−20y′+4y=0, y(5)=0, y′(5)=−e2.
(3 pts) Solve the initial value problem
y′′ − 2√2y′ + 2y = 0, y(√2) = e2, y′(√2) = 2√2e2.
Consider the second order linear equation t2y′′+2ty′−2y=0,
t>0.
(a) (1 pt) Show that y1(t) = t−2 is a solution.
(b) (3 pt) Use the variation of parameters method to obtain a
second solution and a general solution.

Use a Laplace transform to solve the initial value problem: y'''
=y+1,
y(0) = 0,
y'(0) = 0,
y''(0) = 0.

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