February 24, 2021 Exhibit 5.1

Golub Capital BDC, Inc.

200 Park Avenue, 25^th Floor

New York, NY 10166

Ladies and Gentlemen:

We have acted as counsel to Golub Capital BDC, Inc., a Delaware corporation (the “Company”), in connection with the public offering of $400,000,000 in aggregate principal amount of the Company’s 2.500% unsecured notes due 2026 (the “Notes”), pursuant to the Company’s registration statement on Form N-2 (File No. 333-232387) (as amended as of the date hereof, the “Registration Statement”) filed by the Company with the Securities and Exchange Commission (the “Commission”) under the Securities Act of 1933, as amended (the “Securities Act”), which became automatically effective upon filing with the Commission, relating to the public offering of securities of the Company that may be offered by the Company from time to time as set forth in the prospectus dated June 26, 2019, which was included in the Registration Statement, and which forms a part of the Registration Statement (the “Prospectus”), and as may be set forth from time to time in one or more supplements to the Prospectus. This opinion letter is rendered in connection with such public offering of the Notes, as described in the Prospectus and a prospectus supplement dated February 17, 2021 (the “Prospectus Supplement”). All of the Notes are to be sold by the Company as described in the Registration Statement and related Prospectus and Prospectus Supplement.

The Notes will be issued pursuant to the indenture, dated October 2, 2020, entered into between the Company and U.S. Bank National Association, as trustee (the “Trustee”), as supplemented by a second supplemental indenture, dated February 24, 2021, entered into between the Company and the Trustee (collectively, the “Indenture”).

As counsel to the Company, we have reviewed the Registration Statement, the Prospectus and the Prospectus Supplement, and have examined the originals or copies of the following:

(i) The Certificate of Incorporation of the Company, as amended by the Certificate of Amendment of the Certificate of Incorporation of the Company, certified as of a recent date by the Secretary of State of the State of Delaware;

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(ii) The Bylaws of the Company, certified as of the date hereof by an officer of the Company;

(iii) A Certificate of Good Standing with respect to the Company issued by the Secretary of State of the State of Delaware as of a recent date;

(iv) The resolutions of the board of directors, or a duly authorized committee thereof, of the Company relating to, among other things, (a) the authorization and approval of the preparation and filing of the Registration Statement and (b) the authorization, execution and delivery of the Indenture;

(v) the Indenture; and

(vi) a specimen copy of the form of the Notes to be issued pursuant to the Indenture in the form attached to the Indenture.

With respect to such examination and our opinion expressed herein, we have assumed, without any independent investigation or verification, (i) the genuineness of all signatures on all documents submitted to us for examination, (ii) the legal capacity of all natural persons, (iii) the authenticity of all documents submitted to us as originals, (iv) the conformity to original documents of all documents submitted to us as conformed or reproduced copies and the authenticity of the originals of such copied documents, (v) that all certificates issued by public officials have been properly issued, (vi) the accuracy and completeness of all corporate records made available to us by the Company and (vii) that the Indenture will be a valid and legally binding obligation of the parties thereto (other than the Company).

This opinion letter has been prepared, and should be interpreted, in accordance with customary practice followed in the preparation of opinion letters by lawyers who regularly give, and such customary practice followed by lawyers who on behalf of their clients regularly advise opinion recipients regarding, opinion letters of this kind.

As to certain matters of fact relevant to the opinions in this opinion letter, we have relied up certificates and/or representations of officers of the Company. We have also relied on certificates and confirmations of public officials. We have not independently established the facts, or in the case of certificates or confirmations of public officials, the other statements, so relied upon.

The opinion set forth below is limited to the contract laws of the State of New York, as in effect on the date hereof, and we express no opinion with respect to any other laws of the State of New York or the laws of any other jurisdiction. Without limiting the preceding sentence, we express no opinion as to any state securities or broker-dealer laws or regulations thereunder relating to the offer, issuance or sale of the Notes.

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On the basis of and subject to the foregoing, and subject to the all of the assumptions, qualifications and limitations set forth in this opinion letter, we are of the opinion that, when the Notes are duly executed and delivered by duly authorized officers of the Company and duly authenticated by the Trustee, all in accordance with the provisions of the Indenture, and delivered to the purchasers thereof against payment of the agreed consideration therefor, the Notes will constitute valid and legally binding obligations of the Company, enforceable against the Company in accordance with their terms, except as such enforceability may be limited by applicable bankruptcy, insolvency, reorganization, receivership, moratorium, fraudulent conveyance, and other similar laws affecting the rights and remedies of creditors generally and to general principles of equity (including without limitation the availability of specific performance or injunctive relief and the application of concepts of materiality, reasonableness, good faith and fair dealing), regardless of whether considered in a proceeding at law or in equity.

The opinions expressed in this opinion letter (a) are strictly limited to the matters stated in this opinion letter, and without limiting the foregoing, no other opinions are to be inferred and (b) are only as of the date of this opinion letter, and we are under no obligation, and do not undertake, to advise the Company or any other person or entity either of any change of law or fact that occurs, or of any fact that comes to our attention, after the date of this opinion letter, even though such change or such fact may affect the legal analysis or a legal conclusion in this opinion letter.

We hereby consent to the filing of this opinion as an exhibit to the Company’s Current Report on Form 8-K, dated February 24, 2021, and to the reference to our firm in the “Legal Matters” section in the Prospectus Supplement. We do not admit by giving this consent that we are in the category of persons whose consent is required under Section 7 of the Securities Act.

/s/ EVERSHEDS SUTHERLAND (US) LLP