First Order Differential Equations PDF

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Chapter(2(First(Order(Differential(Equations( ( 2.1(Introduction:(Motion(of(a(Falling(Body( ! An!object!falls!through!the!air!toward!Earth.!!Assuming!that!the!only!forces!acting!on! the!object!are!gravity!and!air!resistance,!determine!the!velocity!of!the!object!as!a! function!of!time.! ! Newton’s!second!law:!force!is!equal!to!mass!times!acceleration.!If!F!is!the!total!force,! m!is!the!mass,!and!dv/dt!is!acceleration,!! ! ! F!=! ! When!air!resistance!is!neglected,!we!know!that!F'='mg!where!g!is!the!acceleration! due!to!gravity;!and!air!resistance!is!proportional!to!velocity!and!works!in!an! opposite!direction!from!acceleration.!!The!proportionality!constant!b!depends!on! the!density!of!the!air!and!the!shape!of!the!object.!Therefore,! ! dv m = !!! ! dt ! How!can!this!differential!equation!be!solved,!i.e.!how!can!we!use!this!to!find!a! function!for!the!velocity!v(t)!of!the!object!at!time!t?! ! ! ! ! ! ! ! ! ! !

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! What!is!the!limiting!or!terminal(velocity,!i.e.! lim v(t )?! !!t→∞ ! ! ! 2.2(Separable(Equations( ( A!differential!equation!is!separable!if!it!can!be!written!in!the!form! ! dy = !! dx ! ! Classify!each!as!either!separable!or!not!separable:! ! dy 2x + xy = 2 dx y +1 !!! dy = 1+ xy dx ! dy !Method(to(solve(a(separable(equation:( = g(x)p( y) ( !dx ( ( ( ( ( (

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*Note:(when!dividing!by!a!function!p(y),!any!value!c!such!that!p(c)!=!0!will!result!in!a! constant!function!solution!!y'='c!!which!may!or!may!not!be!included!in!the!solution! found!by!separating!variables.! ! dy x −5 Example!1:!!Solve! = 2 !!!!!!! ! ! dx y ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! (Note!the!location!of!the!constant!C).! ! dy y −1 = Example!2:!!Solve!the!initial!value!problem!! , y(−1) = 0 ! !dx x + 3 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

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dy 6x 5 −2x +1 Example!3:!!Solve!! = ! cos y + e y !!dx ! ! ! ! ! ! ! ! ! ! ! ! dy Example!4:! = 2t + 2 y 2t , y(1) = 0 ! !dt ! ! ! ! ! ! ! ! ! ! ! ! ! 2.2!Group!Exercises:! ! 1.!!Classify!each!as!either!separable!or!not!separable:! ! dy = 4 y 2 − 3 y +1 dx dy ye x+ y = dx x 2 + 2

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ds s +1 = st !!dt ! !

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dy 3 −2 y 2.!!Give!an!explicit!solution!for! = 8x e .!! !dx ! ! ! ! ! ! ! 3.!!Solve!the!initial!value!problem!(give!an!explicit!solution!if!possible):! ! ! ydx +(1 + x )dy = 0, y(0) = 1 !! ! ! ! ! ! ! ! ! ! ! ! ! ! 2.3(Linear(Equations( ( A!linear(first-order(equation!can!be!expressed!in!the!form! ! ! ! ! Where!a0(x),!a1(x)!and!b(x)!depend!only!on!x,!not!on!y.! ! Classify!each!as!either!separable,!linear,!both!or!neither.! ! dy x 2 sin x −(cos x) y = (sin x) dx y

dy +(sin x) y 3 = e x + 1 dx

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dy = cos x dx

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The!first!step!in!solving!a!linear!order!1!equation!is!to!put!it!in!standard(form,!by! dividing!through!by!the!coefficient!of!dy/dx:! ! ! ! ! ! Next,!the!technique!we!use!is!to!multiply!through!by!an!integrating(factor,( !µ (x) ,! that!will!force!the!left!side!of!the!equation!to!be!the!result!of!a!product!rule!(there! are!many!possibilities!for!such!a!factor,!and!we!only!need!to!find!one!that!will!work).! ! Multiply!the!equation!by!the!integrating!factor,! ! ! ! ! ! Thinking!of!the!left!side!as!the!derivative!of!a!product,!this!means!that! ! µ'(x) = !! ! ! ! ! ! ! ! ! Use!separation!of!variables!to!solve!for!!µ(x) = !! ! ! ! ! ! ! dy Since! µ (x) + µ(x)P(x) y = !!! dx ! ! We!must!solve!the!following!DE:! ! ! ! ! ! ! ! ! This!is!called!the!general(solution.! !

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Example!1:!Solve!by!finding!the!integrating!factor,!multiply!through,!write!using! product!rule,!integrate!both!sides,!and!solve!for!y(x).! ! 1 dy 2 y − = x cos x , x > 0!!! x dx x 2 ! ! ! ! ! ! ! ! ! ! ! ! ! *Note:!the!constant!of!integration!is!usually!the!coefficient!of!a!function,!it!is!not!just! a!constant!tacked!on!to!the!end!of!the!solution!! !

! Rather!than!apply!this!process!each!time,!we!will!develop!a!short-cut!method!to! solve!any!linear!order!1!DE:! ! Method(for(Solving(Linear((order(1)(Equations( ! (a)!write!in!standard!form!! ! ! (b)!calculate!an!integrating!factor!(simplify!as!much!as!possible)!! µ(x) = !! ! ! (c)!use!the!formula!! y(x) = !!

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dy + y = x , y(0) = 4 !!! Example!2:!!Solve!the!initial!value!problem:! ! dx !!Transient(terms!approach!0!as!!t → ∞ .!!Identify!any!transient!terms.! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! Existence(and(Uniqueness(of(a(Solution( ! Theorem(1.!!Suppose!P(x)!and!Q(x)!are!continuous!on!an!interval!(a,!b)!that! contains!the!point!x0.!!Then!for!any!choice!of!initial!value!y0,!there!exists!a!unique! solution!to!the!initial!value!problem! dy + P(x) y = Q( x), y( x 0 ) = y 0!.! !dx The!values!of!x!for!which!P(x)!is!not!defined!are!called!singular(points.! dr Example!3:!Find!the!solution! + r secθ = cosθ , r(0) = 1 .!!Give!the!largest!interval!I! !d θ over!which!the!general!solution!is!defined.!!! ! ! ! ! ! ! ! ! ! ! ! ! !

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2.3!Group!Exercises:! ! 1.!!Classify!each!as!separable,!linear,!both!or!neither:! ! dy (t 2 +1) = yt − y dt !! dx x +t 2 x = sint ! dt ! ! 2.!!Find!the!general!solution!for!each.!!Identify!any!transient!terms!and!give!the! largest!interval!of!definition.! dy = x 2e −4 x − 4 y dx

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dx t + 2x = 5t 3 !! dt ! ( ( ( ( ( ( ( 2.4(Exact(Equations( ( A!function!of!two!variables,!!F(x , y) ,!has!partial(derivatives!given!by!!!!

∂F (x , y) = !! !∂x ! ! ∂F (x , y) = ∂y ! !

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Example!1:!Find!the!partial!derivatives!of! !F(x , y) = 4x y + cos x − ln y !! ! ∂F (x , y) = !! !∂x ! ∂F (x , y) = ! ∂ y ! ! Example!2:!!Find!the!second!mixed!partials!for!the!same!function.!!What!do!you! notice?! ! ∂ ∂F (x , y) = !! !∂ y ∂x ! ∂ ∂F (x , y) = ! !∂x ∂ y ! A!level(curve!is!an!equation!!F(x , y) = C ;!for!example,!!4x 5 y 2 + cos x − ln y = 8 .!!This! curve!could!be!graphed!in!the!xy-plane!using!technology.!!We!can!find!a!differential! equation!for!dy/dx!for!this!curve!using!implicit!differentiation:! ! ! ! ! ! ! This!can!be!written!in!form! M(x , y)dx + N(x , y)dy = 0 .! ! ! ! Our!goal!in!this!section!is!to!identify!which!differential!equations!of!the!form! M(x , y)dx + N(x , y)dy = 0 !have!solutions!of!the!form! F(x , y) = C !and!to!use!an! appropriate!method!to!solve!them.!!Can!you!relate!M(x,!y)!and!N(x,!y)!to!F(x,!y)?! A!differential!equation!is!called!exact!in!a!rectangle!R!if!and!only!if!it!can!be!written! in!the!form! ! ! where!M(x,!y)!and!N(x,!y)!are!continuous!in!R!and! ! ! !

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2xy 2 +1 dy Example!3:!!Is!the!given!differential!equation!exact?!! = − !!! 2x 2 y !dx ! ! ! ∂F Can!a!potential(function!!F(x , y) !be!found!such!that! (x , y) = M(x , y) !!and!!! !∂x ∂F (x , y) = N( x , y) ?! !∂ y ! ! ! ! ! ! ! ! Use!the!potential!function!to!give!a!general!(implicit)!solution!to!the!differential! equation;! F(x , y) = C .!

! Terminology:!!!dF( x , y) = M( x , y)dx + N(x , y)dy !is!called!the!total(or!exact( differential!of!F.! ! How!do!we!know!that!a!potential!function! F(x , y) !can!always!be!found,!if!the! equation!is!exact?!The!proof!gives!a!method!for!solving!an!exact!equation.! !

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Method(for(Solving(Exact(Equations( (a)!Show!that!the!equation! M(x , y)dx + N(x , y)dy = 0!is!exact.!!Since!

∂F = M(x , y),!! !∂x

then!!!F(x , y) = !! ! ∂F = N(x , y) ,!take!the!partial!of!F!with!respect!to!y!and!set!equal!to!! (b)!!Since! !∂ y N(x,!y).!!Solve!for!g’(y).! ! (c)!!Integrate!to!find! g( y) !and!substitute!into!the!equation!for!F(x,!y)!in!part!(a).! ! (d)!The!general!implicit!solution!to!the!exact!equation!is! F(x , y) = C ,!where!C!is!a! constant.! ∂F = N(x , y) !to!find!!F(x , y) = ! Note:!you!could!also!begin!with! ∂y Example!4:!!Solve!the!initial!value!problem!! (tan y − 2)dx +(x sec2 y +1/ y)dy = 0, y(0) = 1 !! ! ! ! ! ! ! ! ! ! ! ! ! ! Example!5:!!If!the!equation!is!exact,!find!the!solution! (1 + e x y + xe x y )dx +(xe x + 2)dy = 0.! ! ! ! ! ! ! ! ! ! ! !

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2.4!Group!Exercises! ! 1.!!Classify!as!separable,!linear,!exact,!or!none!of!these.!!Some!may!have!more!than! one!classification.! ! (x 2 y + x 4 cos x)dx − x 3dy = 0

( ye xy + 2x)dx +(xe xy − 2 y)dy = 0 !!

θ dr +(3r − θ −1)dθ = 0 ! ! ! 2.!!If!the!equation!is!exact,!find!the!solution!(2xy + 3)dx +(x 2 −1)dy = 0, y(1) = 2 .! ! ! ! ! ! ! ! !! ! ! 2.6(Substitutions(and(Transformations( ( We!will!cover!only!Homogeneous!and!Bernoulli!equations!using!substitutions.! ! General!technique:! (a)!classify!the!differential!equation!and!use!the!appropriate!substitution! (b)!rewrite!the!equation!using!the!new!variables! (c)!solve!the!transformed!equation! (d)!express!the!solution!in!terms!of!the!original!variables.! ! A.(Homogeneous(Equations( ( An!equation!is!homogeneous!if!it!can!be!written!as!a!function!of!y/x!alone,!i.e.!! ⎛ y⎞ dy = F(x , y) = G ⎜ ⎟ .! dx ⎝ x⎠ !

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Equivalently,!an!equation!in!the!form! M(x , y)dx + N(x , y)dy = 0 !is!homogeneous!if! both!M!and!N!are!homogeneous!functions!of!the!same!degree,!a;!that!is!

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M(tx ,ty ) = t a M(x , y), N(tx ,ty ) = t a N(x , y) !

! Example!1a:!Classify!the!equation!(xy + y 2 + x 2 )dx − x 2dy = 0 !as!exact,!homogeneous,! both!or!neither.! ! ! ! ! ! ! ! ! ! ! ! Method(for(Solving(Homogeneous(Equations( ( y dy (a)!use!the!substitutions! υ = !!and!! = !! x dx ! ! (b)!rewrite!the!equation!using!only!the!variables! x ,υ !and!solve!as!a!separable! equation.! ! (c)!express!the!answer!using!the!original!variables.! ! Example!1b:!Solve!(xy + y 2 + x 2 )dx − x 2dy = 0 .! ! ! ! ! ! ! ! ! ! ! ! ! ! !

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2 2 2 Example!2:!!Solve! (x + y )dx +(x − xy)dy = 0!.! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! B.(Bernoulli(Equations( ( A!first!order!Bernoulli(equation!can!be!written!in!the!form! ! ! ! ! where!P(x)!and!Q(x)!are!continuous!on!an!interval!(a,!b)!and!n!is!a!real!number.! ! Note:!when!n!=!0!or!1,!the!equation!is!also!a!___________________________________!equation.! ! Method(to(Solve(Bernoulli(Equations,(!n ≠ 0,1 ! ! dy (a)!use!the!substitutions! !υ = y 1−n!!and!! = !! dx ! ! ! (b)!rewrite!the!equation!using!only!the!variables! x , υ !and!solve!as!a!linear!equation.! dυ + P1 (x)υ = Q1 (x) !!! ! !dx ! (c)!express!the!answer!using!the!original!variables.! !

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dy 5 Example!3:!!Solve! −5 y = − xy 3 .!! 2 !dx ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! dy Example!4:!!Solve! x + y = x 2 y 2 .! ! dx !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

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2.6!Group!Exercises! ! Show!that!! 2xydy +(x 2 − y 2 )dx = 0!!is!both!homogeneous!and!Bernoulli.! ! ! ! ! ! ! ! ! dυ 1−n + P(x)υ = Q(x)!using!an! Use!the!substitution! !υ = y !to!write!in!the!form! !dx appropriate!Bernoulli!substitution.!!Solve.! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

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Chapter!2!Review! !

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