Practice problems for these concepts can be found at: Applications of Definite Integrals Practice Problems for AP Calculus

### General Procedure

- Separate the variables:
*g*(*y*)*dy*=*f*(*x*)*dx*. - Integrate both sides:
*g*(*y*)*dy*=*f*(*x*)*dx*. - Solve for y to get a general solution.
- Substitute given conditions to get a particular solution.
- Verify your result by differentiating.

### Example 1

Given = 4*x*^{3} *y*^{2} and *y*(1) = , solve the differential equation.

Step 1. Separate the variables: *dy* = 4*x*^{3}*dx*.

Step 2. Integrate both sides:

Step 3. General solution:

Step 4. Particular solution:

Step 5. Verify result by differentiating.

### Example 2

Find a solution of the differentiation equation = *x*sin(*x*^{2}); *y*(0) = – 1.

Step 1. Separate variables: *dy* = *x* sin(*x*^{2})*dx*.

Step 2. Integrate both sides:

Step 3. Substitute given condition:

### Example 3

If = 2*x* +1 and at *x* = 0, *y* ' = – 1, and *y* = 3, find a solution of the differential equation.

Step 1. Rewrite

Step 2. Separate variables: *dy* ' = (2*x* +1)d*x*.

Step 3. Integrate both sides: *dy* ' = (2*x* +1)*dx*; *y* ' = *x*^{2} + *x* + *C*_{1}.

Step 4. Substitute given condition: At *x* =0, *y* ' = – 1; –1 = 0 + 0 + *C*_{1} *C*_{1} = – 1.

- Thus,

*y*' =

*x*

^{2}+

*x*– 1.

Step 5. Rewrite: *dy* ' = = *x*^{2} + *x* – 1.

Step 6. Separate variables: *dy* = (*x*^{2} + *x* – 1)*dx*.

Step 7. Integrate both sides: *dy* = (*x*^{2} + *x* – 1) *dx*

Step 8. Substitute given condition: At *x* = 0, *y* '= 3; 3 = 0 + 0 – 0 + *C*_{2} *C*_{2} = 3.

- Therefore,

Step 9. Verify result by differentiating:

### Example 4

Find the general solution of the differential equation

Step 1. Separate variables:

Step 2. Integrate both sides: (let *u* = *x*^{2} +1; *du* = *2**x* *dx*)

- In|

*y*|=ln(

*x*

^{2}+ 1) +

*C*

_{1}.

Step 3. General Solution: solve for *y*.

*ln|*

_{e}*y*| =

*ln(*

_{e}*x*

^{2}+ 1) +

*C*

_{1}

- |

*y*| = e

^{ln(x2 + 1)}· e

^{C1}; |

*y*| = e

^{C1}(

*x*

^{2}+ 1)

*y*= ±

*e*

^{C1}(

*x*

^{2}+1)

- The general solution is

*y*=

*C*(

*x*

^{2}+ 1).

Step 4. Verify result by differentiating:

*y*=

*C*(

*x*

^{2}+ 1)

### Example 5

Write an equation for the curve that passes through the point (3, 4) and has a slope at any point (x, y) as

Step 1. Separate variables: 2*y* *dy* = (*x*^{2} + 1)*dx*.

Step 2. Integrate both sides:

Step 3. Substitute given condition:

- Thus,

Step 4. Verify the result by differentiating:

Practice problems for these concepts can be found at: Applications of Definite Integrals Practice Problems for AP Calculus

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