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1. Solving Differential Equations

source : intmath.com

1. Solving Differential Equations

A differential equation (or “DE”) contains
derivatives or differentials.

Our task is to solve the differential equation. This will involve integration at some point, and we’ll (mostly) end up with an expression along the lines of “y = …”.

Recall from the Differential section in the Integration chapter, that a differential can be thought of as a derivative where `dy/dx` is actually not written in fraction form.

Examples of Differentials

dx (this means “an infinitely small change in x”)

`d\theta` (this means “an infinitely small change in `\theta`”)

`dt` (this means “an infinitely small change in t”)

Examples of Differential Equations

Example 1

We saw the following example in the Introduction to this chapter. It involves a derivative, `dy/dx`:

`(dy)/(dx)=x^2-3`

As we did before, we will integrate it. This
will be a general solution (involving K, a
constant of integration).

So we proceed as follows:

`y=int(x^2-3)dx`

and this gives

`y=x^3/3-3x+K`

But where did that dy go from the `(dy)/(dx)`? Why did it seem to disappear?

In this example, we appear to be integrating the x part only (on the right), but in fact we have integrated with respect to y as well (on the left). DEs are like that – you need to integrate with respect to two (sometimes more) different variables, one at a time.

We could have written our question only using differentials:

dy = (x2 − 3)dx

(All I did was to multiply both sides of the original dy/dx in the question by dx.)

Now we integrate both sides, the left side with respect to y (that’s why we use “dy”) and the right side with respect to x (that’s why we use “dx”) :

`int dy = int(x^2 – 3)dx`

Then the answer is the same as before, but this time we have arrived at it considering the dy part more carefully:

`y=x^3/3-3x+K`

On the left hand side, we have integrated `int dy = int 1 dy` to give us y.

Note about the constant: We have integrated both sides, but there’s a constant of integration on the right side only. What happened to the one on the left? The answer is quite straightforward. We do actually get a constant on both sides, but we can combine them into one constant (K) which we write on the right hand side.

Example 2

This example also involves differentials:

`\theta^2 d\theta = sin(t + 0.2) dt`

We have:

A function of `theta` with `d theta` on the left side, and

A function of t with dt on the right side.

To solve this, we would integrate both sides, one at a time, as follows:

`int theta^2 d theta = int sin(t+0.2)dt`

`{theta^3}/3 = -cos(t + 0.2) + K`

We have integrated with respect to θ on the left and with respect to t on the right.

Here is the graph of our solution, taking `K=2`:

Typical solution graph for the Example 2 DE: `theta(t)=root(3)(-3cos(t+0.2)+6)`.

Solving a differential equation

From the above examples, we can see that solving a DE means finding
an equation with no derivatives that satisfies the given
DE. Solving a differential equation always involves one or more
integration steps.

It is important to be able to identify the type of
DE we are dealing with before we attempt to
solve it.

Definitions

First order DE: Contains only first derivatives

Second order DE: Contains second derivatives (and
possibly first derivatives also)

Degree: The highest power of the highest
derivative which occurs in the DE.

Example 3 – Order and Degree

a) `(d^2y)/(dx^2)+((dy)/(dx))^3-3x+2y=8`

This DE has order 2 (the highest derivative appearing
is the second derivative) and degree 1 (the
power of the highest derivative is 1.)

b) `((dy)/(dx))^5-2x=3 sin(x)-sin(y)`

This DE has order 1 (the highest derivative appearing
is the first derivative) and degree 5 (the
power of the highest derivative is 5.)

c) `(y”)^4+2(y’)^7-5y=3`

This DE
has order 2 (the highest derivative appearing is the
second derivative) and degree 4 (the power
of the highest derivative is 4.)

General and Particular Solutions

When we first performed integrations, we obtained a general
solution (involving a constant, K).

We obtained a particular solution by substituting known
values for x and y. These known conditions are
called boundary conditions (or initial
conditions).

It is the same concept when solving differential equations – find general solution first, then substitute given numbers to find particular solutions.

Let’s see some examples of first order, first degree DEs.

Example 4

a. Find the general solution for the differential
equation

`dy + 7x dx =
0`

b. Find the particular solution given that `y(0)=3`.

Answer

(a) We simply need to subtract 7x dx from both sides, then insert integral signs and integrate:

`dy=-7x dx`

`intdy=-int7x dx`

`y=-7/2x^2+K`

NOTE 1: We are now writing our (simple) example as a differential equation. Earlier, we would have written this example as a basic integral, like this:

`(dy)/(dx)+7x=0`

Then `(dy)/(dx)=-7x` and so `y=-int7x dx=-7/2x^2+K`

The answer is the same – the way of writing it, and thinking about it, is subtly different.

NOTE 2: `int dy` means `int1 dy`, which gives us the answer `y`.

We could also have:

`intdt=t`

`intd theta=theta`

` int da=a`

and so on. We’ll come across such integrals a lot in this section.

(b) We now use the information y(0) = 3 to find K.

The information means that at x = 0, y = 3. We substitute these values into the equation that we found in part (a), to find the particular solution.

`3=7/2(0)^2+K` gives K = 3.

So the particular solution is: `y=-7/2x^2+3`, an “n”-shaped parabola.

Here is the graph of the particular solution we just found:

Solution graph: `y=-7/2 x^2 + 3`.

Example 5

Find the particular solution of

`y’ = 5`

given that when `x=0, y=2`.

Answer

We can write

y’ = 5

as a differential equation:

dy = 5 dx

Integrating both sides gives:

y = 5x + K

Applying the boundary conditions: x = 0, y = 2, we have K = 2 so:

y = 5x + 2

Example 6

Find the particular solution of

`y”’ = 0`

given that:

`y(0) = 3,` `y'(1) = 4,` `y”(2) = 6`

Answer

Since y”’ = 0, when we integrate once we get:

y” = A (A is a constant)

Integrating again gives:

y’ = Ax + B (A, B are constants)

Once more:

`y = (Ax^2)/2 + Bx + C` (A, B and C are constants)

The boundary conditions are:

y(0) = 3, y'(1) = 4, y”(2) = 6

We need to substitute these values into our expressions for y” and y’ and our general solution, `y = (Ax^2)/2 + Bx + C`.

Now

y(0) = 3 gives C = 3.

and

y” (2) = 6 gives A = 6

(Actually, y” = 6 for any value of x in this problem since there is no x term)

Finally,

y’ (1) = 4 gives B = -2.

So the particular solution for this question is:

y = 3×2 − 2x + 3

Checking the solution by differentiating and substituting initial conditions:

y’ = 6x − 2

y’ (1) = 6(1) − 2 = 4

y” = 6

y”’ = 0

Our solution is correct.

Example 7

After solving the differential
equation,

`(dy)/(dx)ln x-y/x=0`

(we will see how to solve this DE in the next
section Separation of Variables), we obtain the result

`y=c ln x`

Did we get the correct general solution?

Answer

Now, if `y=c ln x`, then `(dy)/(dx)=c/x`

[See Derivative of the Logarithmic Function if you are rusty on this.)

So

`”LHS”=(dy)/(dx)ln x-y/x`

`=(c/x) ln x-((c ln x))/x`

`=0`

`=”RHS”`

We conclude that we have the correct solution.

Second Order DEs

We include two more examples here to give you an idea of second order DEs. We will see later in this chapter how to solve such Second Order Linear DEs.

Example 8

The general solution of the second order DE

y” + a2y = 0

is

`y = A cos ax + B sin ax`

Example 9

The general solution of the second order DE

y” − 3y’ + 2y = 0

is

y = Ae2x + Bex

If we have the following boundary conditions:

y(0) = 4, y'(0) = 5

then the particular solution is given by:

y = e2x + 3ex

Now we do some examples using second order DEs where we are given a final answer and we need to check if it is the correct solution.

Example 10 – Second Order DE

Show that

`y = c_1 sin 2x + 3 cos 2x`

is a general solution for the differential
equation

`(d^2y)/(dx^2)+4y=0`

Answer

We have a second order differential equation and we have been given the general solution. Our job is to show that the solution is correct.

We do this by substituting the answer into the original 2nd order differential equation.

We need to find the second derivative of y:

y = c1 sin 2x + 3 cos
2x

First derivative:

`(dy)/(dx)=2c_1 cos 2x-6 sin 2x`

Second derivative:

`(d^2y)/(dx^2)=-4c_1 sin 2x-12 cos 2x`

Now for the check step:

`”LHS”=(d^2y)/(dx^2)+4y`

`=[-4c_1sin 2x-12 cos 2x]+` `4(c_1sin 2x+3 cos 2x)`

`=0`

`=”RHS”`

Example 11 – Second Order DE

Show that `(d^2y)/(dx^2)=2(dy)/(dx)` has a
solution of y = c1 + c2e2x

Answer

Since

y = c1 + c2e2x, then:

`(dy)/(dx)=2c_2e^(2x)`

and

`(d^2y)/(dx^2)=4c_2e^(2x)`

It is obvious that .`(d^2y)/(dx^2)=2(dy)/(dx)`

Example 9 - Find general solution of dy/dx = x+1 / 2-y - Examples

Example 9 – Find general solution of dy/dx = x+1 / 2-y – Examples – Example 9 – Chapter 9 Class 12 Differential Equations. Last updated at Dec. 24, 2018 by Teachoo. Learn Maths with all NCERT Solutions Class 6 Class 7 Class 8 Class 9 Class 10 Class 11 Class 12.It can first be observed that this differential equation is linear, so we can solve it by multiplying the whole equation by the integrating factor. As there is no coefficient in front of the dy/dx term, we do not have to do anything to the equation before finding the integrating factor.Differential Equation Calculator. The calculator will find the solution of the given ODE: first-order, second-order, nth-order, separable, linear, exact, Bernoulli, homogeneous, or inhomogeneous. Initial conditions are also supported. Show Instructions.

Give the general solution to the Ordinary Differential Equation… – Differential Equations Solution Guide. A Differential Equation is an equation with a function and And using the Wronskian we can now find the particular solution of the differential equation. Differential Equations with unknown multi-variable functions and their partial derivatives are a…What is a solution to the differential equation #dy/dx=-x/y# with the particular solution #y(1)=-sqrt2#?Divide both sides by . In doing so, we force any possible solutions to exist on either or (the "positive" interval in such a situation is usually taken over the "negative" one) because cannot be 0 in order for us to do this.

Give the general solution to the Ordinary Differential Equation...

Differential Equation Calculator – eMathHelp – Connect and share knowledge within a single location that is structured and easy to search. Learn more. Find the general solution of the differential I'm confused on how to separate this equation. Can someone help me get started or provide some hints? Find the general solution of the…Differential equations with separable variables. Linear inhomogeneous differential equations of the 1st order. dx*(x^2 – y^2) – 2*dy*x*y = 0….Solution Of The Given Differential Equation. (x+1)dy/dx+(x+2)y = 2xe^-x Y= Give The Largest Interval I Over Which The General Solution Is Defined. Determine whether there are any transient terms in the general solution. (Enter the transient terms as a comma-separated list; if there are none…

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