Differential equations problems, solved and explained

Recent questions in Differential equations

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asked 2021-09-15

Find transient terms in this general solution to a differential equation, if there are any
\(\displaystyle{y}={\left({x}+{C}\right)}{\left({\frac{{{x}+{2}}}{{{x}-{2}}}}\right)}\)

Differential equations

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asked 2021-09-15

Find if the following first order differential equations seperable, linear, exact, almost exact, homogeneous, or Bernoulli. Rewrite the equation into standard form for the classification it fits.
\(\displaystyle{\left({x}\right)}{\left({e}^{{y}}\right)}{\left(\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}\right)}={e}^{{-{2}{x}}}+{e}^{{{y}-{2}{x}}}\)

Differential equations

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asked 2021-09-13

\(\displaystyle{\left(\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}\right)}=-\frac{{{y}^{{2}}+{x}^{{2}}}}{{{2}{x}{y}}}{\quad\text{and}\quad}{y}{\left({1}\right)}={4}\)
Please, solve the differential equation. Write the method you used and solve for the dependent variable it it is possible.

Differential equations

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asked 2021-09-13

Find if the following first order differential equations seperable, linear, exact, almost exact, homogeneous, or Bernoulli. Rewrite the equation into standard form for the classification it fits.
\(\displaystyle{\left(\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}\right)}={x}^{{2}}{\left[{\left({x}^{{3}}\right)}{\left({y}\right)}-{\left(\frac{{1}}{{x}}\right)}\right]}\)

Differential equations

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asked 2021-09-10

If \(\displaystyle{x}^{{2}}+{x}{y}+{y}^{{3}}={1}\) find the value of y''' at the point where x = 1

Differential equations

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asked 2021-09-09

First using a trigonometric
identity, find \(\displaystyle{L}{\left\lbrace{f{{\left({t}\right)}}}\right\rbrace}\)
\(\displaystyle{f{{\left({t}\right)}}}={\sin{{2}}}{t}{\cos{{2}}}{t}\)

Differential equations

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asked 2021-09-09

Find the length of the curve. \(\displaystyle{r}{\left({t}\right)}={<}{8}{t},{t}^{{2}},{\frac{{{1}}}{{{12}}}}{t}^{{3}}{>},{0}\leq{t}\leq{1}\)

Differential equations

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asked 2021-09-09

If \(\displaystyle{x}{y}+{6}{e}^{{y}}={6}{e}\) , find the value of y" at the point where x=0

Differential equations

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asked 2021-09-09

Solve the differential equation by variation of parameters
\(\displaystyle{y}\text{}+{3}{y}'+{2}{y}={\frac{{{1}}}{{{1}+{e}^{{x}}}}}\)

Differential equations

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asked 2021-09-09

Solve the given differential equation by separation of variables: \(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={x}\sqrt{{{1}-{y}^{{2}}}}\)

Differential equations

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asked 2021-09-07

Explain what is the difference between implicit and explicit solutions for differential equation initial value problems.

Differential equations

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asked 2021-09-04

\(\displaystyle{\left({x}-{y}\right)}{\left.{d}{x}\right.}+{x}{\left.{d}{y}\right.}={0}\)
Please, solve by using an appropriate substitution.

Differential equations

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asked 2021-09-03

Find if the following first order differential equations seperable, linear, exact, almost exact, homogeneous, or Bernoulli. Rewrite the equation into standard form for the classification it fits.
\(\displaystyle{\left[{\left({2}\right)}{\left({x}\right)}{\left({y}^{{3}}\right)}+{1}\right]}{\left.{d}{x}\right.}={\left[{\left({y}^{{-{{1}}}}\right)}-{\left({3}\right)}{\left({x}^{{2}}\right)}{\left({y}^{{2}}\right)}\right]}{\left.{d}{y}\right.}\)

Differential equations

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asked 2021-09-03

Find if the following first order differential equations seperable, linear, exact, almost exact, homogeneous, or Bernoulli. Rewrite the equation into standard form for the classification it fits.
\(\displaystyle-{\left[{y}^{{4}}-{\left({t}\right)}{\left({y}^{{3}}\right)}\right]}{\left.{d}{t}\right.}={\left({3}\right)}{\left({t}^{{2}}\right)}{\left({y}^{{2}}\right)}{\left.{d}{y}\right.}\)

Differential equations

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asked 2021-07-05

Transform the second-order differential equation \(\displaystyle{\frac{{{d}^{{{2}}}{x}}}{{{d}{t}^{{{2}}}}}}={3}{x}\) into a system of first-order differential equations.

Differential equations

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asked 2021-07-05

True or False? Justify your answer with a proof or a counterexample. You can determine the behavior of all first-order differential equations using directional fields or Euler's method.

Differential equations

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asked 2021-06-16

Use the substitution \(y'=v\) to write each second-order equation as a system of two first-order differential equations (planar system). \(\displaystyle{y}^{''}+\mu{\left({t}^{{{2}}}-{1}\right)}{y}^{''}+{y}={0}\)

Differential equations

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asked 2021-06-12

Suppose that a population develops according to the logistic equation
\(\frac{dP}{dt}=0.05P-0.0005P^2\)
where t is measured in weeks. What is the carrying capacity? What is the value of k?

Differential equations

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asked 2021-06-11

If \(x^2 + xy + y^3 = 1\) find the value of y''' at the point where x = 1

Differential equations

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asked 2021-06-07

Ask your question. Get your answer. Easy as that

Find transient terms in this general solution to a differential equation, if there are any
\(\displaystyle{y}={\left({x}+{C}\right)}{\left({\frac{{{x}+{2}}}{{{x}-{2}}}}\right)}\)

\(\displaystyle{\left(\frac{{\left.{d}{y}\right.}}{{\left.{d}{x}\right.}}\right)}=-\frac{{{y}^{{2}}+{x}^{{2}}}}{{{2}{x}{y}}}{\quad\text{and}\quad}{y}{\left({1}\right)}={4}\)
Please, solve the differential equation. Write the method you used and solve for the dependent variable it it is possible.

If \(\displaystyle{x}^{{2}}+{x}{y}+{y}^{{3}}={1}\) find the value of y''' at the point where x = 1

First using a trigonometric
identity, find \(\displaystyle{L}{\left\lbrace{f{{\left({t}\right)}}}\right\rbrace}\)
\(\displaystyle{f{{\left({t}\right)}}}={\sin{{2}}}{t}{\cos{{2}}}{t}\)

Find the length of the curve. \(\displaystyle{r}{\left({t}\right)}={<}{8}{t},{t}^{{2}},{\frac{{{1}}}{{{12}}}}{t}^{{3}}{>},{0}\leq{t}\leq{1}\)

If \(\displaystyle{x}{y}+{6}{e}^{{y}}={6}{e}\) , find the value of y" at the point where x=0

Solve the differential equation by variation of parameters
\(\displaystyle{y}\text{}+{3}{y}'+{2}{y}={\frac{{{1}}}{{{1}+{e}^{{x}}}}}\)

Solve the given differential equation by separation of variables: \(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={x}\sqrt{{{1}-{y}^{{2}}}}\)

Explain what is the difference between implicit and explicit solutions for differential equation initial value problems.

\(\displaystyle{\left({x}-{y}\right)}{\left.{d}{x}\right.}+{x}{\left.{d}{y}\right.}={0}\)
Please, solve by using an appropriate substitution.

Transform the second-order differential equation \(\displaystyle{\frac{{{d}^{{{2}}}{x}}}{{{d}{t}^{{{2}}}}}}={3}{x}\) into a system of first-order differential equations.

True or False? Justify your answer with a proof or a counterexample. You can determine the behavior of all first-order differential equations using directional fields or Euler's method.

Use the substitution \(y'=v\) to write each second-order equation as a system of two first-order differential equations (planar system). \(\displaystyle{y}^{''}+\mu{\left({t}^{{{2}}}-{1}\right)}{y}^{''}+{y}={0}\)

Suppose that a population develops according to the logistic equation
\(\frac{dP}{dt}=0.05P-0.0005P^2\)
where t is measured in weeks. What is the carrying capacity? What is the value of k?

If \(x^2 + xy + y^3 = 1\) find the value of y''' at the point where x = 1

Find \(\frac{dy}{dx}\) using implicit differentiation \(xe^{y}=x-y\)

Ask your question. Get your answer. Easy as that

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