MATH 106D - CALCULUS II
WINTER 2005
QUIZ 8
NAME:
Show ALL your work CAREFULLY.
Consider the initial value problem
dy
dx =x2y, y(0) = 1.
(a) Use the technique of separation of variables to find the solution of this
differential equation.
By separating the variables, we have
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ydy =x2dx.
It follows that
ln |y|=x3
3+C
⇒y=Kex3/3.
Since y(0) = 1,K=1and hence the solution is y=ex3/3.
(b) Use Euler’s method to estimate y(0.75) with initial point (0,1) and
stepsize ∆x=0.25.
The n-thstepintheEuler’smethodyieldsthey-coordinate as
yn=f(xn−1,y
n−1)·∆x+yn−1
where the differential equation is dy
dx =f(x, y).Heref(x, y )=x2y.
y1=f(0,1) ·1
4+1=1;
y2=f1
4,1·1
4+1= 65
64;(now x2is 1
2)
y3=f1
2,65
64·1
4+65
64 =1105
1024 ;
⇒y(0.75) = 1105
1024 .
Date: March 25, 2005.
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