"Find the solution of the following differential equations:..." solved and explained

hunterofdeath63 Answered2021-12-30

Find the solution of the following differential equations: Homogeneous Differential Equations

З х у^{2} d x + (3 x^{2} у + у^{3}) d y = 0; y (1) = 2

(x^{2} - 9 y^{2}) d x + 2 x y d y = 0; y (2) = - 1

Answer & Explanation

Matthew Rodriguez

Expert

2021-12-31Added 32 answers

The given homogenous differential equation

3 x y^{2} d x + (3 x^{2} y + y^{3}) d y = 0

(1)

y (1) = 2

(2)
First we check the Exactness of the given Differential equation,

M d x + N d y = 0

∴ M = 3 x y^{2}, N = 3 x^{2} y + y^{3}

Then,

\frac{\partial M}{\partial y} = 6 x y, \frac{\partial N}{\partial x} = 6 x y

∴ \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}

So, the given differential equation is exact.
Hence the solution is given by.

\int M d x + \int (terms in N not containg x) d y = c

\int 3 x y^{2} d x + \int y^{3} d y = c

3 y^{2} \cdot \frac{x^{2}}{2} + \frac{y^{4}}{4} = c

(3)

\frac{3}{2} x^{2} y^{2} + \frac{y^{4}}{4} = c

using (2)
Put

x = 1, y = 2

3 {(1)}^{2} {(2)}^{2} + \frac{{(2)}^{4}}{4} = c

6 + 4 = c \Rightarrow c = 10

(3) becomes

\frac{3}{2} x^{2} y^{2} + \frac{y^{4}}{4} = 10

3 x^{2} y^{2} + y^{4} = 40

This is the required solution.

einfachmoipf

Expert

2022-01-01Added 32 answers

(x^{2} - 9 y^{2}) d x + 2 x y d y = 0

y (2) = - 1

(x^{2} - 9 y^{2}) d x = - 2 x y d y

\frac{d y}{d x} = \frac{- (x^{2} - 9 y^{2})}{2 x y}

y = v x

\frac{d y}{d x} = v + x \frac{d v}{d x}

v + x \frac{d v}{d x} = \frac{- (x^{2} - 9 v^{2} x^{2})}{2 \times x \times v x}

v + x \frac{d v}{d x} = \frac{- x^{2} (1 - 9 v^{2})}{2 x^{2} \times v} = \frac{- (1 - 9 v^{2})}{2 v}

x \frac{d v}{d x} = \frac{- 1 + 9 v^{2} - 2 v^{2}}{2 v} = \frac{- 1 + 7 v^{2}}{2 v}

\int \frac{2 v}{7 v^{2} - 1} d v = \int \frac{d x}{x}

Let

7 v^{2} - 1 = f

14 \cdot b \cdot d v = d t \Rightarrow 2 v d v = \frac{d t}{7}

\frac{1}{7} \int \frac{d t}{t} = \int \frac{d t}{x}

\frac{1}{7} \ln (t) = \ln (x) + \ln c

{\ln (7 v^{2} - 1)}^{\frac{1}{7}} = \ln (x \cdot c)

This is helpful 0 Flag

Vasquez

Expert

2022-01-09Added 457 answers

$\begin{matrix} (1) 3 x y^{2} d x + (3 x^{2} y + y^{3}) d y = 0 (1) \\ y (1) = 2 (2) \\ First we check the Exactness of the given Differential e q u a t i o n, \\ M d x + N d y = 0 \\ ∴ M = 3 x y^{2}, N = 3 x^{2} y + y^{3} \\ T h e n, \frac{\partial M}{\partial y} = 6 x y, \frac{\partial N}{\partial x} = 6 x y \\ ∴ \frac{\partial M}{\partial y} = This is helpful 0 FlagDo you have a similar question?Recalculate according to your conditions! RecalculateAsk your question. Get your answer.Let our experts help you. Answer in as fast as 15 minutes. Ask a QuestionMost Popular Questions Previous talpajocotefnf3 2022-03-27 For what values of x is f (x) = 2 x^{4} + 4 x^{3} + 2 x^{2} - 2 concave or convex? See answers (1) dotzis16xd 2022-03-24 Let's say I have two samples of results of two bernoulli experiments. H_{0} : p_{1} = p_{2} H_{1} : p_{1} \neq p_{2} And I want to try to reject H_{0} at a confidence level. I already know a proper way to solve this, but I was wondering, if I have a confidence interval for p_{1} and p_{2}, at the same level of significance. Can I just check if the intervals overlaps each other to test this? See answers (2) Litzy Wallace 2022-03-27 Define whether sequence is arithmetic or geometric and write the n-th term formula 1) 11, 17, 23, \dots 2) 5, 15, 45, \dots See answers (1) Aryanna Rowland 2022-03-16 What is the instantaneous rate of change of f (x) = (x^{2} - 3 x) e^{x} at x=2? See answers (1) Rylee Friedman 2023-01-08 Which three-dimensional figure has half the number of vertices as the rectangular prism? See answers (1) talpajocotefnf3 2022-03-27 For what values of x is f (x) = 2 x^{4} + 4 x^{3} + 2 x^{2} - 2 concave or convex? See answers (1) dotzis16xd 2022-03-24 Let's say I have two samples of results of two bernoulli experiments. H_{0} : p_{1} = p_{2} H_{1} : p_{1} \neq p_{2} And I want to try to reject H_{0} at a confidence level. I already know a proper way to solve this, but I was wondering, if I have a confidence interval for p_{1} and p_{2}, at the same level of significance. Can I just check if the intervals overlaps each other to test this? See answers (2) Litzy Wallace 2022-03-27 Define whether sequence is arithmetic or geometric and write the n-th term formula 1) 11, 17, 23, \dots 2) 5, 15, 45, \dots See answers (1) Aryanna Rowland 2022-03-16 What is the instantaneous rate of change of f (x) = (x^{2} - 3 x) e^{x} at x=2? See answers (1) Rylee Friedman 2023-01-08 Which three-dimensional figure has half the number of vertices as the rectangular prism? See answers (1) talpajocotefnf3 2022-03-27 For what values of x is f (x) = 2 x^{4} + 4 x^{3} + 2 x^{2} - 2 concave or convex? See answers (1) dotzis16xd 2022-03-24 Let's say I have two samples of results of two bernoulli experiments. H_{0} : p_{1} = p_{2} H_{1} : p_{1} \neq p_{2} And I want to try to reject H_{0} at a confidence level. I already know a proper way to solve this, but I was wondering, if I have a confidence interval for p_{1} and p_{2}, at the same level of significance. Can I just check if the intervals overlaps each other to test this? See answers (2) Litzy Wallace 2022-03-27 Define whether sequence is arithmetic or geometric and write the n-th term formula 1) 11, 17, 23, \dots 2) 5, 15, 45, \dots See answers (1) Next 12345Didn't find what you were looking for?Product High School Questions College Questions Textbook Math Solver Company About us Term of Service Privacy Payment Policy Honor Connect with us Contact us Youtube Facebook Get Plainmath App Google Play Louki Akrita, 23, Bellapais Court, Flat/Office 46, 1100, Nicosia, Cyprus Cyprus reg.number: ΗΕ 419361 E-mail us: support@plainmath.net Our Service is useful for: Plainmath is a platform aimed to help users to understand how to solve math problems by providing accumulated knowledge on different topics and accessible examples. Plainmath.net is owned and operated by RADIOPLUS EXPERTS LTD. 2022 Plainmath. All rights reserved Secured by PayPalWelcome back Get answers within minutes and finish your homework faster function statusChangeCallback(response) {
if (response.status === 'connected') {
fbLogin(response);
} else {
console.log('Please log into this webpage.');
}
}
function login() {
document.querySelector('.js-facebook-button').classList.add('loading')
FB.login(function(response) {
statusChangeCallback(response);
}, {scope: 'public_profile,email'});
}
window.fbAsyncInit = function() {
FB.init({
appId : '300431328256302',
cookie : true,
xfbml : true,
version : 'v12.0'
});
};
function fbLogin(response) {
FB.api('/me', function() {
let xmlHttp = new XMLHttpRequest();
xmlHttp.open("GET", '/connect/facebook/check/popup?token=' + response.authResponse.accessToken, true);
xmlHttp.responseType = 'json';
xmlHttp.send();
xmlHttp.onload = function(e) {
if (this.status === 200) {
document.dispatchEvent(new CustomEvent('loginPopupSuccess', {detail: this.response}))
} else if (this.status === 400) {
document.dispatchEvent(new CustomEvent('loginPopupError', {detail: this.response}))
} else {
console.log(e)
}
};
});
} Join with Facebook Sign in with Apple async function apple_func_login_on_popup()
{
try {
const data = await AppleID.auth.signIn();

if (data.authorization.code) {
fetch('/connect/apple/check/popup?code=' + data.authorization.code, {
method: 'GET'
}).then((response) => {
if (response.ok) {
return response.json();
}

return Promise.reject(response);

}).then((response) => {
document.dispatchEvent(new CustomEvent('loginPopupSuccess', {detail: response}))
}).catch((response) => {
console.log('Apple signIn error', response);
});
}
// Handle successful response.
} catch ( error ) {
console.log(error);
// Handle error.
}
} Or Login with email Dont have an account? Register Welcome back Get answers within minutes and finish your homework faster Log In Forgot your password? Dont have an account? Register Join Plainmath.net Create a free account to see answers function statusChangeCallback(response) {
if (response.status === 'connected') {
fbLogin(response);
} else {
console.log('Please log into this webpage.');
}
}
function login() {
document.querySelector('.js-facebook-button').classList.add('loading')
FB.login(function(response) {
statusChangeCallback(response);
}, {scope: 'public_profile,email'});
}
window.fbAsyncInit = function() {
FB.init({
appId : '300431328256302',
cookie : true,
xfbml : true,
version : 'v12.0'
});
};
function fbLogin(response) {
FB.api('/me', function() {
let xmlHttp = new XMLHttpRequest();
xmlHttp.open("GET", '/connect/facebook/check/popup?token=' + response.authResponse.accessToken, true);
xmlHttp.responseType = 'json';
xmlHttp.send();
xmlHttp.onload = function(e) {
if (this.status === 200) {
document.dispatchEvent(new CustomEvent('loginPopupSuccess', {detail: this.response}))
} else if (this.status === 400) {
document.dispatchEvent(new CustomEvent('loginPopupError', {detail: this.response}))
} else {
console.log(e)
}
};
});
} Join with Facebook Sign up with Apple async function apple_func_login_on_popup()
{
try {
const data = await AppleID.auth.signIn();

if (data.authorization.code) {
fetch('/connect/apple/check/popup?code=' + data.authorization.code, {
method: 'GET'
}).then((response) => {
if (response.ok) {
return response.json();
}

return Promise.reject(response);

}).then((response) => {
document.dispatchEvent(new CustomEvent('loginPopupSuccess', {detail: response}))
}).catch((response) => {
console.log('Apple signIn error', response);
});
}
// Handle successful response.
} catch ( error ) {
console.log(error);
// Handle error.
}
} Or Sign up with email Already have an account? Sign infunction onLoadGoogleClient() {
google.accounts.id.initialize({
client_id: GOOGLE_CLIENT_ID,
callback: (response) => {
var xmlHttp = new XMLHttpRequest()
xmlHttp.open("GET", GOOGLE_AUTH_POPUP_PATH + '?credential=' + response.credential)
xmlHttp.responseType = 'json'
xmlHttp.send()
xmlHttp.onload = function(e) {
if (this.status === 200) {
document.dispatchEvent(new CustomEvent('loginPopupSuccess', {detail: this.response}))
} else {
console.log(e)
}
}
}
})
google.accounts.id.prompt()
}

function onLoadAppleClient() {
AppleID.auth.init({
clientId : 'plainmath.prod.service',
scope : 'name email',
redirectURI : 'https://plainmath.net/connect/apple/check',
usePopup : true
});
}(() => { setTimeout(function() { let _13ac8551 = document.createElement('script');_13ac8551.src = '/build/general.js?16751733065509a4f700f';document.body.append(_13ac8551); }, 3000) })() \end{matrix}$