Keep in mind that there is a key pitfall to this method. Consider the differential equation
Based on the form of
we guess a particular solution of the form
But when we substitute this expression into the differential equation to find a value for
we run into a problem. We have
and
so we want
which is not possible.
Looking closely, we see that, in this case, the general solution to the complementary equation is
The exponential function in
is actually a solution to the complementary equation, so, as we just saw, all the terms on the left side of the equation cancel out. We can still use the method of undetermined coefficients in this case, but we have to alter our guess by multiplying it by
Using the new guess,
we have
and
Substitution gives
So,
and
This gives us the following general solution
Note that if
were also a solution to the complementary equation, we would have to multiply by
again, and we would try
Problem-solving strategy: method of undetermined coefficients
Solve the complementary equation and write down the general solution.
Based on the form of
make an initial guess for
Check whether any term in the guess for
is a solution to the complementary equation. If so, multiply the guess by
Repeat this step until there are no terms in
that solve the complementary equation.
Substitute
into the differential equation and equate like terms to find values for the unknown coefficients in
Add the general solution to the complementary equation and the particular solution you just found to obtain the general solution to the nonhomogeneous equation.
Solving nonhomogeneous equations
Find the general solutions to the following differential equations.
The complementary equation is
which has the general solution
(step 1). Based on the form of
our initial guess for the particular solution is
(step 2). None of the terms in
solve the complementary equation, so this is a valid guess (step 3).
Now we want to find values for
and
so substitute
into the differential equation. We have
so we want to find values of
and
such that
Therefore,
This gives
and
so
(step 4).
Putting everything together, we have the general solution
The complementary equation is
which has the general solution
(step 1). Based on the form
our initial guess for the particular solution is
(step 2). However, we see that this guess solves the complementary equation, so we must multiply by
which gives a new guess:
(step 3). Checking this new guess, we see that it, too, solves the complementary equation, so we must multiply by
t again, which gives
(step 3 again). Now, checking this guess, we see that
does not solve the complementary equation, so this is a valid guess (step 3 yet again).
We now want to find a value for
so we substitute
into the differential equation. We have
and
Substituting into the differential equation, we want to find a value of
so that
This gives
so
(step 4). Putting everything together, we have the general solution
The complementary equation is
which has the general solution
(step 1). Based on the form
our initial guess for the particular solution is
(step 2). None of the terms in
solve the complementary equation, so this is a valid guess (step 3). We now want to find values for
and
so we substitute
into the differential equation. We have
and
so we want to find values of
and
such that
Therefore,
This gives
and
so
(step 4). Putting everything together, we have the general solution
The complementary equation is
which has the general solution
(step 1). Based on the form
our initial guess for the particular solution is
(step 2). However, we see that the constant term in this guess solves the complementary equation, so we must multiply by
which gives a new guess:
(step 3). Checking this new guess, we see that none of the terms in
solve the complementary equation, so this is a valid guess (step 3 again). We now want to find values for
and
so we substitute
into the differential equation. We have
and
so we want to find values of
and
such that
Therefore,
This gives
and
so
(step 4). Putting everything together, we have the general solution
Communication is effective because it allows individuals to share ideas, thoughts, and information with others.
effective communication can lead to improved outcomes in various settings, including personal relationships, business environments, and educational settings. By communicating effectively, individuals can negotiate effectively, solve problems collaboratively, and work towards common goals.
it starts up serve and return practice/assessments.it helps find voice talking therapy also assessments through relaxed conversation.
miss
Every time someone flushes a toilet in the apartment building, the person begins to jumb back automatically after hearing the flush, before the water temperature changes. Identify the types of learning, if it is classical conditioning identify the NS, UCS, CS and CR. If it is operant conditioning, identify the type of consequence positive reinforcement, negative reinforcement or punishment
nature is an hereditary factor while nurture is an environmental factor which constitute an individual personality. so if an individual's parent has a deviant behavior and was also brought up in an deviant environment, observation of the behavior and the inborn trait we make the individual deviant.
Samuel
I am taking this course because I am hoping that I could somehow learn more about my chosen field of interest and due to the fact that being a PsyD really ignites my passion as an individual the more I hope to learn about developing and literally explore the complexity of my critical thinking skills
hello. autism is a umbrella term. autistic kids have different disorder overlapping. for example. a kid may show symptoms of ADHD and also learning disabilities.
before treatment please make sure the kid doesn't have physical disabilities like hearing..vision..speech problem. sometimes these
Jharna
continue..
sometimes due to these physical problems..the diagnosis may be misdiagnosed.
treatment for autism.
well it depends on the severity.
since autistic kids have problems in communicating and adopting to the environment.. it's best to expose the child in situations where the child
Jharna
child interact with other kids under doc supervision.
play therapy.
speech therapy.
Engaging in different activities that activate most parts of the brain.. like drawing..painting. matching color board game.
string and beads game.
the more you interact with the child the more effective
Jharna
results you'll get..
please consult a therapist to know what suits best on your child.
and last as a parent. I know sometimes it's overwhelming to guide a special kid.
but trust the process and be strong and patient as a parent.
Jharna
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