1
Solved example of separable differential equations
$\frac{dy}{dx}=\frac{2x}{3y^2}$
2
Rewrite the differential equation in the standard form $M(x,y)dx+N(x,y)dy=0$
$3y^2dy-2xdx=0$
3
The differential equation $3y^2dy-2xdx=0$ is exact, since it is written in the standard form $M(x,y)dx+N(x,y)dy=0$, where $M(x,y)$ and $N(x,y)$ are the partial derivatives of a two-variable function $f(x,y)$ and they satisfy the test for exactness: $\displaystyle\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$. In other words, their second partial derivatives are equal. The general solution of the differential equation is of the form $f(x,y)=C$
$\frac{dy}{dx}=\frac{2x}{3y^2}$
Intermediate steps
Find the derivative of $M(x,y)$ with respect to $y$
$\frac{d}{dy}\left(-2x\right)$
The derivative of the constant function ($-2x$) is equal to zero
0
Find the derivative of $N(x,y)$ with respect to $x$
$\frac{d}{dx}\left(3y^2\right)$
The derivative of the constant function ($3y^2$) is equal to zero
0
4
Using the test for exactness, we check that the differential equation is exact
$0=0$
Intermediate steps
The integral of a function times a constant ($-2$) is equal to the constant times the integral of the function
$-2\int xdx$
Applying the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, in this case $n=1$
$-x^2$
Since $y$ is treated as a constant, we add a function of $y$ as constant of integration
$-x^2+g(y)$
5
Integrate $M(x,y)$ with respect to $x$ to get
$-x^2+g(y)$
Intermediate steps
The derivative of the constant function ($-x^2$) is equal to zero
0
The derivative of $g(y)$ is $g'(y)$
$0+g'(y)$
6
Now take the partial derivative of $-x^2$ with respect to $y$ to get
$0+g'(y)$
Intermediate steps
Simplify and isolate $g'(y)$
$3y^2=0+g$
$x+0=x$, where $x$ is any expression
$3y^2=g$
Rearrange the equation
$g=3y^2$
7
Set $3y^2$ and $0+g'(y)$ equal to each other and isolate $g'(y)$
$g'(y)=3y^2$
Intermediate steps
Integrate both sides with respect to $y$
$g=\int3y^2dy$
The integral of a function times a constant ($3$) is equal to the constant times the integral of the function
$g=3\int y^2dy$
Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, such as $2$
$g=y^{3}$
8
Find $g(y)$ integrating both sides
$g(y)=y^{3}$
9
We have found our $f(x,y)$ and it equals
$f(x,y)=-x^2+y^{3}$
10
Then, the solution to the differential equation is
$-x^2+y^{3}=C_0$
Intermediate steps
We need to isolate the dependent variable $y$, we can do that by subtracting $-x^2$ from both sides of the equation
$y^{3}=x^2+C_0$
Removing the variable's exponent
$y=\sqrt[3]{x^2+C_0}$
11
Find the explicit solution to the differential equation
$y=\sqrt[3]{x^2+C_0}$
Final Answer
$y=\sqrt[3]{x^2+C_0}$
All Differential Equations Resources
Find the general solution to the given system.
Correct answer:
Explanation:
To find the general solution to the given system
first find the eigenvalues and eigenvectors.
Therefore the eigenvalues are
Now calculate the eigenvectors
For
Thus,
For
Thus
Therefore,
Now the general solution is,
Solve the initial value problem . Where
Correct answer:
Explanation:
To solve the homogeneous system, we will need a fundamental matrix. Specifically, it will help to get the matrix exponential. To do this, we will diagonalize the matrix. First, we will find the eigenvalues which we can do by calculating the determinant of .
Finding the eigenspaces, for lambda = 1, we have
Adding -1/2 Row 1 to Row 2 and dividing by -1/2, we have which means
Thus, we have an eigenvector of .
For lambda = 4
Adding Row 1 to Row 2, we have
So with an eigenvector .
Thus, we have and . Using the inverse formula for 2x2 matrices, we have that . As we know that , we have
The solution to a homogenous system of linear equations is simply to multiply the matrix exponential by the intial condition. For other fundamental matrices, the matrix inverse is needed as well.
Thus, our final answer is
Solve the homogenous equation:
With the initial conditions:
Correct answer:
Explanation:
So this is a homogenous, second order differential equation. In order to solve this we need to solve for the roots of the equation. This equation can be written as:
Which, using the quadratic formula or factoring gives us roots of and
The solution of homogenous equations is written in the form:
so we don't know the constants, but can substitute the values we solved for the roots:
We have two initial values, one for y(t) and one for y'(t), both with t=0\
So:
so:
We can solve for : Then plug into the other equation to solve for
So, solving, we get: Then
This gives a final answer of:
Solve the second order differential equation:
Subject to the initial values:
Correct answer:
Explanation:
So this is a homogenous, second order differential equation. In order to solve this we need to solve for the roots of the equation. This equation can be written as:
Which, using the quadratic formula or factoring gives us roots of and
The solution of homogenous equations is written in the form:
so we don't know the constants, but can substitute the values we solved for the roots:
We have two initial values, one for y(t) and one for y'(t), both with t=0
So:
so:
We can solve Then plug into the other equation to solve for
So, solving, we get: Then
This gives a final answer of:
Solve the differential equation for y:
Subject to the initial condition:
Correct answer:
Explanation:
So this is a homogenous, first order differential equation. In order to solve this we need to solve for the roots of the equation. This equation can be written as:
gives us a root of
The solution of homogenous equations is written in the form:
so we don't know the constant, but can substitute the values we solved for the root:
We have one initial values, for y(t) with t=0
So:
This gives a final answer of:
Solve the third order differential equation:
Correct answer:
Explanation:
So this is a homogenous, third order differential equation. In order to solve this we need to solve for the roots of the equation. This equation can be written as:
Which, using the cubic formula or factoring gives us roots of , and
The solution of homogenous equations is written in the form:
so we don't know the constants, but can substitute the values we solved for the roots:
We have three initial values, one for y(t), one for y'(t), and for y''(t) all with t=0
So:
so:
So this can be solved either by substitution or by setting up a 3X3 matrix and reducing. Once you do either of these methods, the values for the constants will be: Then and
This gives a final answer of:
Solve the differential equation:
Subject to the initial conditions:
Correct answer:
Explanation:
So this is a homogenous, third order differential equation. In order to solve this we need to solve for the roots of the equation. This equation can be written as:
Which, using the cubic formula or factoring gives us roots of , and
The solution of homogenous equations is written in the form:
so we don't know the constants, but can substitute the values we solved for the roots:
We have three initial values, one for y(t), one for y'(t), and for y''(t) all with t=0
So:
So this can be solved either by substitution or by setting up a 3X3 matrix and reducing. Once you do either of these methods, the values for the constants will be: Then and
This gives a final answer of:
Find the general solution to the given system.
Correct answer:
Explanation:
To find the general solution to the given system
first find the eigenvalues and eigenvectors.
Therefore the eigenvalues are
Now calculate the eigenvectors
For
Thus,
For
Thus
Therefore,
Now the general solution is,
When substituted into the homogeneous linear system for , which of the following matrices will have a saddle point equilibrium in its phase plane?
Correct answer:
Explanation:
A saddle point phase plane results from two real eigenvalues of different signs. Three of these matrices are triangular, which means their eigenvalues are on the diagonal. For these three, the eigenvalues are real, but both the same sign, meaning they don't have saddles. For the remaining two, we'll need to find the eigenvalues using the characteristic equations.
For , we have
The discriminant to this is , so the solutions are non-real. Thus, this matrix doesn't yield a saddle point.
For we have,
We see that this matrix yields two real eigenvalues with different signs. Thus, it is the correct choice.
Find the general solution to the system of ordinary differential equations
where
Possible Answers:
None of the other answers.
Correct answer:
Explanation:
Finding the eigenvalues and eigenvectors of with the characteristic equation of the matrix
The corresponding eigenvalues are, respectively
and
This gives us that the general solution is