{"id":194925,"date":"2025-02-25T09:31:32","date_gmt":"2025-02-25T09:31:32","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=194925"},"modified":"2025-02-25T09:31:35","modified_gmt":"2025-02-25T09:31:35","slug":"by-inspection-find-a-particular-solution-of-y-2y-14","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/02\/25\/by-inspection-find-a-particular-solution-of-y-2y-14\/","title":{"rendered":"By inspection, find a particular solution of y” + 2y = 14"},"content":{"rendered":"\n

By inspection, find a particular solution of y” + 2y = 14. Yp(x)-<\/p>\n\n\n\n

The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n

A particular solution ( Y_p(x) ) for the differential equation:
[
y” + 2y = 14
]
can be found by inspection.<\/p>\n\n\n\n

Step 1: Identify the Non-Homogeneous Term<\/h3>\n\n\n\n

The right-hand side of the equation is a constant ( 14 ). This suggests trying a constant solution for ( Y_p(x) ).<\/p>\n\n\n\n

Step 2: Assume a Particular Solution Form<\/h3>\n\n\n\n

We assume a constant solution:
[
Y_p = C
]
where ( C ) is a constant to be determined.<\/p>\n\n\n\n

Step 3: Compute Derivatives<\/h3>\n\n\n\n

Since ( Y_p = C ) is a constant, its first and second derivatives are:
[
Y_p’ = 0, \\quad Y_p” = 0.
]<\/p>\n\n\n\n

Step 4: Substitute into the Differential Equation<\/h3>\n\n\n\n

Substituting into the given equation:
[
0 + 2(C) = 14.
]<\/p>\n\n\n\n

Step 5: Solve for ( C )<\/h3>\n\n\n\n

[
2C = 14
]
[
C = 7.
]<\/p>\n\n\n\n

Conclusion: Particular Solution<\/h3>\n\n\n\n

Thus, a particular solution to the equation is:
[
Y_p(x) = 7.
]<\/p>\n\n\n\n

\n\n\n\n

Explanation (300 words)<\/h3>\n\n\n\n

To find a particular solution of ( y” + 2y = 14 ), we analyze the non-homogeneous term. The function on the right-hand side is a constant, which suggests trying a constant function as a particular solution.<\/p>\n\n\n\n

If ( y_p(x) = C ), then the derivatives are zero: ( y_p’ = 0 ) and ( y_p” = 0 ). Substituting into the equation:
[
0 + 2C = 14.
]
Solving for ( C ), we divide by 2 to get ( C = 7 ). This means the function ( Y_p(x) = 7 ) satisfies the equation.<\/p>\n\n\n\n

This method works because when the non-homogeneous term is a constant, the differential equation effectively balances the constant with the coefficient of ( y ). Since the equation contains ( 2y ), solving for ( y ) directly gives ( y = 7 ).<\/p>\n\n\n\n

A key point in this approach is recognizing the right-hand function’s form and choosing an appropriate trial solution. If the right-hand term were more complex (e.g., polynomial, exponential, or trigonometric), we would use different techniques like undetermined coefficients or variation of parameters.<\/p>\n\n\n\n

Finding a particular solution is essential in solving non-homogeneous differential equations because it helps construct the general solution, which consists of the homogeneous solution plus this particular solution.<\/p>\n\n\n\n

Now, let\u2019s generate a visual representation of this process.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Here is a visual representation of the particular solution ( Y_p(x) = 7 ). The red horizontal line at ( y = 7 ) illustrates that the solution is constant for all values of ( x ), as expected from our calculations. Let me know if you need further clarification!<\/p>\n","protected":false},"excerpt":{"rendered":"

By inspection, find a particular solution of y” + 2y = 14. Yp(x)- The Correct Answer and Explanation is : A particular solution ( Y_p(x) ) for the differential equation:[y” + 2y = 14]can be found by inspection. Step 1: Identify the Non-Homogeneous Term The right-hand side of the equation is a constant ( 14 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-194925","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/194925","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=194925"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/194925\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=194925"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=194925"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=194925"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}