Math Problems Vault

Prove that the particular integral of the equation of the previous problem, satisfying the condition $y=1$ at $x=0$ , is expressed by the equation $x^{2}y^{2}+x^{2}+y^{2}=1$ .

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Problem 4037 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 10.10 "Euler Equations"

On the arc of the lemniscate $r^{2}=a^{2}\cos 2\varphi$ , consider points $M(\varphi ,r)$ and $M(\varphi_{1},r_{1})$ such that the arcs $OM$ and $AM_{1}$ are equal, where $O\left(\frac{\pi }{4},0\right)$ , $A(0,a)$ . Prove that $\cos \varphi \cos \varphi_{1}=\dfrac {1}{\sqrt{2}}$ .

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Problem 4038 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 10.10 "Euler Equations"

The equation $\dfrac {dx}{\sqrt{1-x^{2}}}+\dfrac {dy}{\sqrt{1-y^{2}}}=0$ has the general integral $\arcsin x+\arcsin y=C$ . Establish that it also has the algebraic integral

x\sqrt{1-y^{2}}+y\sqrt{1-x^{2}}=C_{1}.

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Problem 4039 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 10.10 "Euler Equations"

Setting $x=\sin \varphi$ , $y=\sin \psi$ , $\sqrt{1-x^{2}}=\cos \varphi$ , $\sqrt{1-y^{2}}=\cos \psi$ , use the result of the previous problem to prove the addition theorem for sines:

\sin \varphi \cos \psi +\sin \psi \cos \varphi =\sin (\varphi +\psi ).

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Problem 4040 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 10.10 "Euler Equations"

In studying the equation

\frac{dx}{\sqrt{(1-x^{2})(1-k^{2}x^{2})}}+\frac{dy}{\sqrt{(1-y^{2})(1-k^{2}y^{2})}}=0

set $\sqrt{(1-x^{2})(1-k^{2}x^{2})}=R(x)$ , $dx=R(x)\, dt$ , $dy=-R(y)\, dt$ , and prove the equalities

yx'-xy'=yR(x)+xR(y),

yx'+xy'=yR(x)-xR(y),

yx''-xy''=2k^{2}xy(x^{2}-y^{2}).

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Problem 4041 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 10.10 "Euler Equations"

Prove that the general integral of the equation of the previous problem can be written in the form

y\sqrt{(1-x^{2})(1-k^{2}x^{2})}+x\sqrt{(1-y^{2})(1-k^{2}y^{2})}=C(1-k^{2}x^{2}y^{2}).

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Problem 4042 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 10.10 "Euler Equations"

Defining the function $x=\operatorname {sn}u$ by the equality

u=\int _{0}^{x}\frac{dx}{\sqrt{(1-x^{2})(1-k^{2}x^{2})}}

and introducing the functions $\sqrt{1-x^{2}}=\operatorname {cn}u$ , $\sqrt{1-k^{2}x^{2}}=\operatorname {dn}u$ , prove the equality

\frac{\operatorname {sn}u\operatorname {cn}v\operatorname {dn}v+\operatorname {sn}v\operatorname {cn}u\operatorname {dn}u}{1-k^{2}\operatorname {sn}^{2}u\operatorname {sn}^{2}v}=\operatorname {sn}(u+v).

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Problem 4043 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 10.10 "Euler Equations"

Prove that the general integral of the equation

\frac{dx}{\sqrt{x(1-x)(1-\lambda x)}}+\frac{dy}{\sqrt{y(1-y)(1-\lambda y)}}=0

can be represented in the form

\sqrt{y(1-x)(1-\lambda x)}+\sqrt{x(1-y)(1-\lambda y)}=C(1-\lambda xy).

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Problem 4044 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 10.10 "Euler Equations"

Prove that the general integral of the equation

\frac{d\varphi }{\sqrt{\cos \varphi }}+\frac{d\psi }{\sqrt{\cos \psi }}=0

can be written in the form

\sqrt{\cos \varphi }-\sqrt{\cos \psi }=C\sin \frac{\varphi -\psi }{2}.

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Euler's equation dx/sqrt(R(x)) + dy/sqrt(R(y)) = 0 and addition theorems

References