Write an equivalent first-order differential equation and initial condition for y.

What is the equivalent first-order differential equation?

What is the initial condition?

b2sonicxh
2022-01-21
Answered

Write an equivalent first-order differential equation and initial condition for y.

What is the equivalent first-order differential equation?

What is the initial condition?

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asked 2022-09-11

What is a solution to the differential equation $\frac{dy}{dx}=\frac{y}{x}$?

asked 2022-05-21

My Problem is this given System of differential Equations:

$\dot{x}=8x+18y$

$\dot{y}=-3x-7y$

I am looking for a gerenal solution.

My Approach was: i can see this is a System of linear and ordinary differential equations. Both are of first-order, because the highest derivative is the first. But now i am stuck, i have no idea how to solve it. A Transformation into a Matrix should lead to this expression:

$\overrightarrow{y}=\left(\begin{array}{cc}8& 18\\ -3& -7\end{array}\right)\cdot x$

or is this correct:

$\overrightarrow{x}=\left(\begin{array}{cc}8& 18\\ -3& -7\end{array}\right)\cdot y\text{?}$

But i don't know how to determine the solution, from this point on.

$\dot{x}=8x+18y$

$\dot{y}=-3x-7y$

I am looking for a gerenal solution.

My Approach was: i can see this is a System of linear and ordinary differential equations. Both are of first-order, because the highest derivative is the first. But now i am stuck, i have no idea how to solve it. A Transformation into a Matrix should lead to this expression:

$\overrightarrow{y}=\left(\begin{array}{cc}8& 18\\ -3& -7\end{array}\right)\cdot x$

or is this correct:

$\overrightarrow{x}=\left(\begin{array}{cc}8& 18\\ -3& -7\end{array}\right)\cdot y\text{?}$

But i don't know how to determine the solution, from this point on.

asked 2021-09-20

Evaluate the indefinite integral as an infinite series. $\int \frac{\mathrm{cos}x-1}{x}dx$

asked 2021-01-08

Solve differential equation${y}^{\prime}-2y=1-2x$

asked 2022-07-02

I have the following differential equation

$s(1-s)t={f}^{\prime}(t)(f(t)-st)$

Initial condition: $f(0)=0.$

I solved it and got the solution as

$f(t)=\sqrt{2{c}_{1}+{s}^{2}-2(s-1)s\mathrm{ln}(t)}+s$

But the answer given is

$f(t)=k(s)t,$

where

$k(s)=\frac{s+\sqrt{4s-3{s}^{2}}}{2}.$

If anyone can provide me some hint on how to proceed and reach the specified answer, I would be really grateful.

$s(1-s)t={f}^{\prime}(t)(f(t)-st)$

Initial condition: $f(0)=0.$

I solved it and got the solution as

$f(t)=\sqrt{2{c}_{1}+{s}^{2}-2(s-1)s\mathrm{ln}(t)}+s$

But the answer given is

$f(t)=k(s)t,$

where

$k(s)=\frac{s+\sqrt{4s-3{s}^{2}}}{2}.$

If anyone can provide me some hint on how to proceed and reach the specified answer, I would be really grateful.

asked 2022-05-21

$g(x)$ is continuous on $[1,2]$ such that $g(1)=0$ and

${g}^{\prime}(x){x}^{2}=\sqrt{1-{g}^{2}(x)}$

Find $g(2)$

I found that $g(x)=\mathrm{sin}(c-\frac{1}{x})$ and since $g(1)=0$ shouldn't my c be equal to 1, so that $\mathrm{sin}(1-1)=0$.

But when I try for $g(2)$ with $c=1$, I am getting a different answer from the textbook.

${g}^{\prime}(x){x}^{2}=\sqrt{1-{g}^{2}(x)}$

Find $g(2)$

I found that $g(x)=\mathrm{sin}(c-\frac{1}{x})$ and since $g(1)=0$ shouldn't my c be equal to 1, so that $\mathrm{sin}(1-1)=0$.

But when I try for $g(2)$ with $c=1$, I am getting a different answer from the textbook.

asked 2022-07-04

How to solve the differential equation $(dy/dx{)}^{2}=(x-y{)}^{2}$ with initial condition $y(0)=0$?

I solved the equation by partitioning it into two differential equations.

1) $dy/dx=x-y$ The solution is $\to $ $1-x+y=-\mathrm{exp}(-x)$

and

2) $1+x-y=\mathrm{exp}(x)$

How do we write combined solution of such equations.

I solved the equation by partitioning it into two differential equations.

1) $dy/dx=x-y$ The solution is $\to $ $1-x+y=-\mathrm{exp}(-x)$

and

2) $1+x-y=\mathrm{exp}(x)$

How do we write combined solution of such equations.