1. Differential Equations
Verification Examples
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

2. Verify that the given function satisfies the differential equation.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

3. Verify that the given function satisfies the differential equation.
First compute the derivative:
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

4.  This is true, so the diff. eq. is satisfied
Verify that the given function satisfies the differential equation.
First compute the derivative:
Now substitute into the differential equation:
 This is true, so the diff. eq. is satisfied
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

5. Verify that the given function satisfies the differential equation,
and find a value for C that satisfies the given initial condition.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

6. Verify that the given function satisfies the differential equation,
and find a value for C that satisfies the given initial condition.
Find the derivative:
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

7. Verify that the given function satisfies the differential equation,
and find a value for C that satisfies the given initial condition.
Find the derivative:
Substitute into diff. eq.:
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

8.  It all cancels out, so the equation is satisfied
Verify that the given function satisfies the differential equation,
and find a value for C that satisfies the given initial condition.
Find the derivative:
Substitute into diff. eq.:
 It all cancels out, so the equation is satisfied
Now we need to satisfy the initial condition.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

9.  It all cancels out, so the equation is satisfied
Verify that the given function satisfies the differential equation,
and find a value for C that satisfies the given initial condition.
Find the derivative:
Substitute into diff. eq.:
 It all cancels out, so the equation is satisfied
Now we need to satisfy the initial condition.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

10.  It all cancels out, so the equation is satisfied
Verify that the given function satisfies the differential equation,
and find a value for C that satisfies the given initial condition.
Find the derivative:
Substitute into diff. eq.:
 It all cancels out, so the equation is satisfied
Now we need to satisfy the initial condition.
Set this equal to -1
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

11.  It all cancels out, so the equation is satisfied
Verify that the given function satisfies the differential equation,
and find a value for C that satisfies the given initial condition.
Find the derivative:
Substitute into diff. eq.:
 It all cancels out, so the equation is satisfied
Now we need to satisfy the initial condition.
Set this equal to -1
Our final solution is
We will call this a particular solution because it matches with a particular initial condition.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB