Calculus Examples

Solve the Differential Equation (dy)/(dx)=7x^6-4x^3+12 , y(1)=24
,
Step 1
Rewrite the equation.
Step 2
Integrate both sides.
Tap for more steps...
Step 2.1
Set up an integral on each side.
Step 2.2
Apply the constant rule.
Step 2.3
Integrate the right side.
Tap for more steps...
Step 2.3.1
Split the single integral into multiple integrals.
Step 2.3.2
Since is constant with respect to , move out of the integral.
Step 2.3.3
By the Power Rule, the integral of with respect to is .
Step 2.3.4
Since is constant with respect to , move out of the integral.
Step 2.3.5
By the Power Rule, the integral of with respect to is .
Step 2.3.6
Apply the constant rule.
Step 2.3.7
Simplify.
Tap for more steps...
Step 2.3.7.1
Simplify.
Tap for more steps...
Step 2.3.7.1.1
Combine and .
Step 2.3.7.1.2
Combine and .
Step 2.3.7.2
Simplify.
Step 2.4
Group the constant of integration on the right side as .
Step 3
Use the initial condition to find the value of by substituting for and for in .
Step 4
Solve for .
Tap for more steps...
Step 4.1
Rewrite the equation as .
Step 4.2
Simplify .
Tap for more steps...
Step 4.2.1
Simplify each term.
Tap for more steps...
Step 4.2.1.1
One to any power is one.
Step 4.2.1.2
One to any power is one.
Step 4.2.1.3
Multiply by .
Step 4.2.1.4
Multiply by .
Step 4.2.2
Simplify by subtracting numbers.
Tap for more steps...
Step 4.2.2.1
Subtract from .
Step 4.2.2.2
Add and .
Step 4.3
Move all terms not containing to the right side of the equation.
Tap for more steps...
Step 4.3.1
Subtract from both sides of the equation.
Step 4.3.2
Subtract from .
Step 5
Substitute for in and simplify.
Tap for more steps...
Step 5.1
Substitute for .