+0

# Differential equation

0
388
3

Determine the solution to the differential equation 2y '+ y = x2 passing through the point (-1; -3).
i know the result, but didn't understand the way it's been solved

the result should be

y=-9,7e-0,5x+x2-4x+8

Thx!

Guest Aug 27, 2015

#2
+26547
+5

Here's my solution:

It should be checked (it's different from your suggested result).

.

Alan  Aug 30, 2015
Sort:

#1
0

Determine the solution to the differential equation 2y '+ y = x2 passing through the point (-1; -3).

I hope you can go from here!.

y(x) = c_1 e^(-x/2)+x^2-4 x+8

or:y'(x) = x^2/2-(y(x))/2

Guest Aug 27, 2015
#2
+26547
+5

Here's my solution:

It should be checked (it's different from your suggested result).

.

Alan  Aug 30, 2015
#3
0

The original answer is correct. The mistake in the answer above is at the point where the -1 and -3 are substituted to find the value of k. The 1, in brackets on the rhs, should be -1.

Here's an alternative method of solution.

The solution consists of a Complementary Function (CF) which is the general solution of the homogeneous equation 2dy/dx + y = 0, plus a Particular Integral (PI), which is any solution of the complete equation 2dy/dx + y = x^2.

The CF is easily found to be A.exp(-x/2), where A is an arbitrary constant.

There are several methods for calculating a PI, easiest is probably by trial solution.

Let  $$y=ax^2+bx+c,$$ so  $$y'=2ax+b$$.

Substitute into the differential equation, and we have

$$2(2ax+b)+(ax^2+bx+c)\equiv x^2$$,

or,   $$ax^2+(4a+b)x+(2b+c)\equiv x^2.$$

Now equate coefficients across the identity.

$$a=1,$$

$$4a+b=0,$$ so,  $$b=-4,$$

$$2b+c=0,$$ so, $$c=8.$$

That means the general solution of the original ode is   $$y = A\exp(-x/2)+x^2-4x+8,$$

and substituting the -1 and -3 gets $$A = -16 \exp(-1/2)\approx -9.70449.$$

Guest Sep 1, 2015

### 28 Online Users

We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details