FIRST-ORDER ORDINARY DIFFERENTIAL EQUATIONS
14.28 Find the solution of
(5x+y−7)dy
dx=3(x+y+1).14.29 Find the solutiony=y(x)of
xdy
dx+y−y^2
x^3 /^2=0,
subject toy(1) = 1.
14.30 Find the solution of
(2 siny−x)dy
dx=tany,if (a)y(0) = 0, and (b)y(0) =π/2.
14.31 Find the family of solutions of
d^2 y
dx^2+
(
dy
dx) 2
+
dy
dx=0
that satisfyy(0) = 0.14.5 Hints and answers14.1 N(t)=N 0 exp(−λt).
14.3 (a) exact,x^2 y^4 +x^2 +y^2 =c;(b)IF=x−^1 /^2 ,x^1 /^2 (x+y)=c;(c)IF=
sec^2 x, y^2 tanx+y=c.
14.5 (a) IF = (1−x^2 )−^2 ,y=(1−x^2 )(k+sin−^1 x); (b) IF = cosecx, leading to
y=ksinx+cosx; (c) exact equation isy−^1 (dx/dy)−xy−^2 =y, leading to
x=y(k+y^2 /2).
14.7 y(t)=e−t/α
∫t
α−^1 et′/α
f(t′)dt′;(a)y(t)=1−e−t/α;(b)y(t)=α−^1 e−t/α;(c)y(t)=
(e−t/α−e−t/β)/(α−β). It becomes case (b).
14.9 Note that, if the angle between the tangent and the radius vector isα,then
cosα=dr/dsand sinα=p/r.
14.11 Homogeneous equation, puty=vxto obtain (1−v)(v^2 +2v+2)−^1 dv=x−^1 dx;
write 1−vas 2−(1 +v), andv^2 +2v+2 as 1+(1+v)^2 ;
A[x^2 +(x+y)^2 ]=exp
{
4tan−^1 [(x+y)/x]}
.
14.13 (1 +s)(d ̄y/ds)+2 ̄y=0.C= 1; use separation of variables to show directly that
y(t)=te−t.
14.15 The equation is of the form of (14.22), setv=x+y;x+3y+2ln(x+y−2) =A.
14.17 The equation is isobaric with weighty=−2; settingy=vx−^2 gives
v−^1 (1−v)−^1 (1− 2 v)dv=x−^1 dx;4xy(1−x^2 y)=1.
14.19 The curve must satisfyy=(1−p−^1 )−^1 (1−x+px), which has solutionx=(p−1)−^2 ,
leading toy=(1±
√
x)^2 orx=(1±√
y)^2 ; the singular solutionp′= 0 gives
straight lines joining (θ,0) and (0, 1 −θ)foranyθ.
14.21 v=qu+q/(q−1), whereq=dv/du. General solutiony^2 =cx^2 +c/(c−1),
hyperbolae forc>0 and ellipses forc<0. Singular solutiony=±(x±1).
14.23 (a) Integrating factor is (a^2 +x^2 )^1 /^2 ,y=(a^2 +x^2 )/3+A(a^2 +x^2 )−^1 /^2 ; (b) separable,
y=x(x^2 +Ax+4)−^1.
14.25 Use Laplace transforms;xs ̄(s^2 +4)=s+s^2 − 2 e−^3 s;
x(t)=^12 sin 2t+cos2t−^12 H(t−3) +^12 cos(2t−6)H(t−3).
14.27 This is Clairaut’s equation withF(p)=A/p.Generalsolutiony=cx+A/c;
singular solution,y=2
√
Ax.
14.29 Either Bernoulli’s equation withn= 2 or an isobaric equation withm=3/2;
y(x)=5x^3 /^2 /(2 + 3x^5 /^2 ).