5 Steps to a 5 AP Calculus BC 2019

322 STEP 4. Review the Knowledge You Need to Score High

Example 2

Find a solution of the differentiation equation

dy

dx

=xsin(x^2 );y(0)=−1.

Step 1. Separate variables:dy=xsin(x^2 )dx.

Step 2. Integrate both sides:

∫

dy=

∫

xsin(x^2 )dx;

∫

dy=y.

Letu=x^2 ;du= 2 xdxor

du

2

=xdx.

∫

xsin(x^2 )dx=

∫

sinu

(

du

2

)

=

1

2

∫

sinudu=−

1

2

cosu+C

=−

1

2

cos(x^2 )+C

Thus,y=−

1

2

cos(x^2 )+C.

Step 3. Substitute given condition:

y(0)=−1;− 1 =−

1

2

cos(0)+C;− 1 =

− 1

2

+C;−

1

2

=C.

Thus,y=−

1

2

cos(x^2 )−

1

2

.

Step 4. Verify the result by differentiating:

dy

dx

=

1

2

[

sin(x^2 )

]

(2x)=xsin(x^2 ).

Example 3

If

d^2 y

dx^2

= 2 x+1 and atx=0,y′=−1, andy=3, find a solution of the differential equation.

Step 1. Rewrite

d^2 y

dx^2

as

dy′

dx

;

dy′

dx

= 2 x+1.

Step 2. Separate variables:dy′=(2x+1)dx.

Step 3. Integrate both sides:

∫

dy′=

∫

(2x+1)dx;y′=x^2 +x+C 1.

Step 4. Substitute given condition: Atx=0,y′=−1;− 1 = 0 + 0 +C 1 ⇒C 1 =−1. Thus,

y′=x^2 +x−1.

Step 5. Rewrite:y′=

dy

dx

;

dy

dx

=x^2 +x−1.