2nd Order Differentials Summary
Academic Skills Advice
A 2nd order differential equation is one that has a 2nd derivative in it. For example:
To solve this equation you would need to find , which is a function of . This summary
demonstrates a method to solve the equation.
Summary of steps to solve:
Write the Auxillary Equation (AE)
Solve AE to find the Complementary Function (CF)
Find the Particular Integral (PI)
Find the complete general solution
Find the Particular Solution (if applicable)
Particular Integral
e.g.
e.g.
CF + PI
X
Some things you’ll need to know:
How you write the Complementary Function will depend on the roots of the Auxiliary
Equation that you find:
Type of roots:
Real & different
Real & equal
Complex (
Complementary Function:
)
Special cases of the Complementary Function:
There is a quick solution for the CF if the equation has no
Equation:
term:
Complementary Function
Useful general forms for the Right Hand Side:
If:
Assume:
n.b. If the general form of the RHS is already included in the CF then multiply the assumed
general form by , then continue as before.
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Examples:
1. If the RHS of the equation is 0, then you only need to find the CF – that will be the final
answer.
e.g. Solve the differential equation
Step 1: Use the coefficients to write the AE:
Step 2: Solve AE to find CF:
Real & different roots
so we use the 1st form
of the CF.
Complementary Function:
If you were given initial conditions you could also find A and B but if not this is the final
answer to the question.
2. If the RHS is a function then you need to do all the steps and find the Complete Solution.
e.g. Solve the differential equation
, given that when
Step 1: Use the coefficients to write the AE:
Step 2: Solve AE to find CF:
Real & different roots
so we use the 1st form
of the CF.
Complementary Function:
This time there is a function on the right hand side (RHS) so we continue on to find the PI.
We need to decide which general form of the RHS to use (by looking at the table above).
Step 3: Find the PI
Looking at the RHS,
, so (from the table) we assume the general form:
Differentiate to get:
Now substitute the above into the original equation:
Original:
Becomes:
Tidy up to get:
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Equate Coefficients to find
:
:
:
Nos:
Therefore, Particular Integral
Step 4: Write the complete general solution:
The complete General Solution is CF+PI
So we have:
Step 5: Find the particular solution:
In this question we have been given the initial conditions (when
we can continue on to find A and B and write the particular solution.
) so
So far we know that:
Therefore:
Substitute initial conditions into the above:
When
When
:
:
Solve simultaneously to find that,
and
We now have the particular solution:
© H Jackson 2013 / Academic Skills
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e.g. Solve the differential equation
Real & different roots
so we use the 1st form
of the CF.
Step 1: Use the coefficients to write the AE:
Step 2: Solve AE to find CF:
Complementary Function:
Step 3: Find the PI
Looking at the RHS,
, so we assume the general form:
Differentiate to get:
The trick here is to simplify and say:
First substitute
Next substitute
into the original equation:
and
Equate coefficients of
Equate coefficients of
in:
:
:
Solve these equations simultaneously to find:
and
Particular Integral
Step 4: Write the complete general solution:
The complete General Solution is CF+PI
So we have:
n.b. if we had been given initial conditions we could now find A and B.
© H Jackson 2013 / Academic Skills
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